analysis.complex.operator_norm
⟷
Mathlib.Analysis.Complex.OperatorNorm
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Analysis.Complex.Basic
-import Analysis.NormedSpace.OperatorNorm
+import Analysis.NormedSpace.OperatorNorm.Basic
import Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"33c67ae661dd8988516ff7f247b0be3018cdd952"
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -47,7 +47,7 @@ theorem reCLM_norm : ‖reCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖reCLM 1‖ := by simp
- _ ≤ ‖reCLM‖ := unit_le_op_norm _ _ (by simp)
+ _ ≤ ‖reCLM‖ := unit_le_opNorm _ _ (by simp)
#align complex.re_clm_norm Complex.reCLM_norm
-/
@@ -64,7 +64,7 @@ theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖imCLM I‖ := by simp
- _ ≤ ‖imCLM‖ := unit_le_op_norm _ _ (by simp)
+ _ ≤ ‖imCLM‖ := unit_le_opNorm _ _ (by simp)
#align complex.im_clm_norm Complex.imCLM_norm
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -25,82 +25,82 @@ open ContinuousLinearMap
namespace Complex
-#print Complex.det_conjLie /-
+#print Complex.det_conjLIE /-
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
-theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
+theorem det_conjLIE : (conjLIE.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
det_conjAe
-#align complex.det_conj_lie Complex.det_conjLie
+#align complex.det_conj_lie Complex.det_conjLIE
-/
-#print Complex.linearEquiv_det_conjLie /-
+#print Complex.linearEquiv_det_conjLIE /-
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
-theorem linearEquiv_det_conjLie : conjLie.toLinearEquiv.det = -1 :=
+theorem linearEquiv_det_conjLIE : conjLIE.toLinearEquiv.det = -1 :=
linearEquiv_det_conjAe
-#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLie
+#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLIE
-/
-#print Complex.reClm_norm /-
+#print Complex.reCLM_norm /-
@[simp]
-theorem reClm_norm : ‖reClm‖ = 1 :=
+theorem reCLM_norm : ‖reCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
- 1 = ‖reClm 1‖ := by simp
- _ ≤ ‖reClm‖ := unit_le_op_norm _ _ (by simp)
-#align complex.re_clm_norm Complex.reClm_norm
+ 1 = ‖reCLM 1‖ := by simp
+ _ ≤ ‖reCLM‖ := unit_le_op_norm _ _ (by simp)
+#align complex.re_clm_norm Complex.reCLM_norm
-/
-#print Complex.reClm_nnnorm /-
+#print Complex.reCLM_nnnorm /-
@[simp]
-theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
- Subtype.ext reClm_norm
-#align complex.re_clm_nnnorm Complex.reClm_nnnorm
+theorem reCLM_nnnorm : ‖reCLM‖₊ = 1 :=
+ Subtype.ext reCLM_norm
+#align complex.re_clm_nnnorm Complex.reCLM_nnnorm
-/
-#print Complex.imClm_norm /-
+#print Complex.imCLM_norm /-
@[simp]
-theorem imClm_norm : ‖imClm‖ = 1 :=
+theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
- 1 = ‖imClm I‖ := by simp
- _ ≤ ‖imClm‖ := unit_le_op_norm _ _ (by simp)
-#align complex.im_clm_norm Complex.imClm_norm
+ 1 = ‖imCLM I‖ := by simp
+ _ ≤ ‖imCLM‖ := unit_le_op_norm _ _ (by simp)
+#align complex.im_clm_norm Complex.imCLM_norm
-/
-#print Complex.imClm_nnnorm /-
+#print Complex.imCLM_nnnorm /-
@[simp]
-theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
- Subtype.ext imClm_norm
-#align complex.im_clm_nnnorm Complex.imClm_nnnorm
+theorem imCLM_nnnorm : ‖imCLM‖₊ = 1 :=
+ Subtype.ext imCLM_norm
+#align complex.im_clm_nnnorm Complex.imCLM_nnnorm
-/
-#print Complex.conjCle_norm /-
+#print Complex.conjCLE_norm /-
@[simp]
-theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
- conjLie.toLinearIsometry.norm_toContinuousLinearMap
-#align complex.conj_cle_norm Complex.conjCle_norm
+theorem conjCLE_norm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖ = 1 :=
+ conjLIE.toLinearIsometry.norm_toContinuousLinearMap
+#align complex.conj_cle_norm Complex.conjCLE_norm
-/
-#print Complex.conjCle_nnorm /-
+#print Complex.conjCLE_nnorm /-
@[simp]
-theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
- Subtype.ext conjCle_norm
-#align complex.conj_cle_nnorm Complex.conjCle_nnorm
+theorem conjCLE_nnorm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
+ Subtype.ext conjCLE_norm
+#align complex.conj_cle_nnorm Complex.conjCLE_nnorm
-/
-#print Complex.ofRealClm_norm /-
+#print Complex.ofRealCLM_norm /-
@[simp]
-theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
- ofRealLi.norm_toContinuousLinearMap
-#align complex.of_real_clm_norm Complex.ofRealClm_norm
+theorem ofRealCLM_norm : ‖ofRealCLM‖ = 1 :=
+ ofRealLI.norm_toContinuousLinearMap
+#align complex.of_real_clm_norm Complex.ofRealCLM_norm
-/
-#print Complex.ofRealClm_nnnorm /-
+#print Complex.ofRealCLM_nnnorm /-
@[simp]
-theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
- Subtype.ext <| ofRealClm_norm
-#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnorm
+theorem ofRealCLM_nnnorm : ‖ofRealCLM‖₊ = 1 :=
+ Subtype.ext <| ofRealCLM_norm
+#align complex.of_real_clm_nnnorm Complex.ofRealCLM_nnnorm
-/
end Complex
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
-import Mathbin.Analysis.Complex.Basic
-import Mathbin.Analysis.NormedSpace.OperatorNorm
-import Mathbin.Data.Complex.Determinant
+import Analysis.Complex.Basic
+import Analysis.NormedSpace.OperatorNorm
+import Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"33c67ae661dd8988516ff7f247b0be3018cdd952"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
Copyright (c) Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.complex.operator_norm
-! leanprover-community/mathlib commit 33c67ae661dd8988516ff7f247b0be3018cdd952
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Analysis.Complex.Basic
import Mathbin.Analysis.NormedSpace.OperatorNorm
import Mathbin.Data.Complex.Determinant
+#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"33c67ae661dd8988516ff7f247b0be3018cdd952"
+
/-! # The basic continuous linear maps associated to `ℂ`
> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -28,18 +28,23 @@ open ContinuousLinearMap
namespace Complex
+#print Complex.det_conjLie /-
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
det_conjAe
#align complex.det_conj_lie Complex.det_conjLie
+-/
+#print Complex.linearEquiv_det_conjLie /-
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
theorem linearEquiv_det_conjLie : conjLie.toLinearEquiv.det = -1 :=
linearEquiv_det_conjAe
#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLie
+-/
+#print Complex.reClm_norm /-
@[simp]
theorem reClm_norm : ‖reClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -47,12 +52,16 @@ theorem reClm_norm : ‖reClm‖ = 1 :=
1 = ‖reClm 1‖ := by simp
_ ≤ ‖reClm‖ := unit_le_op_norm _ _ (by simp)
#align complex.re_clm_norm Complex.reClm_norm
+-/
+#print Complex.reClm_nnnorm /-
@[simp]
theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
Subtype.ext reClm_norm
#align complex.re_clm_nnnorm Complex.reClm_nnnorm
+-/
+#print Complex.imClm_norm /-
@[simp]
theorem imClm_norm : ‖imClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -60,31 +69,42 @@ theorem imClm_norm : ‖imClm‖ = 1 :=
1 = ‖imClm I‖ := by simp
_ ≤ ‖imClm‖ := unit_le_op_norm _ _ (by simp)
#align complex.im_clm_norm Complex.imClm_norm
+-/
+#print Complex.imClm_nnnorm /-
@[simp]
theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
Subtype.ext imClm_norm
#align complex.im_clm_nnnorm Complex.imClm_nnnorm
+-/
+#print Complex.conjCle_norm /-
@[simp]
theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
conjLie.toLinearIsometry.norm_toContinuousLinearMap
#align complex.conj_cle_norm Complex.conjCle_norm
+-/
+#print Complex.conjCle_nnorm /-
@[simp]
theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
Subtype.ext conjCle_norm
#align complex.conj_cle_nnorm Complex.conjCle_nnorm
+-/
+#print Complex.ofRealClm_norm /-
@[simp]
theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
ofRealLi.norm_toContinuousLinearMap
#align complex.of_real_clm_norm Complex.ofRealClm_norm
+-/
+#print Complex.ofRealClm_nnnorm /-
@[simp]
theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
Subtype.ext <| ofRealClm_norm
#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnorm
+-/
end Complex
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -46,7 +46,6 @@ theorem reClm_norm : ‖reClm‖ = 1 :=
calc
1 = ‖reClm 1‖ := by simp
_ ≤ ‖reClm‖ := unit_le_op_norm _ _ (by simp)
-
#align complex.re_clm_norm Complex.reClm_norm
@[simp]
@@ -60,7 +59,6 @@ theorem imClm_norm : ‖imClm‖ = 1 :=
calc
1 = ‖imClm I‖ := by simp
_ ≤ ‖imClm‖ := unit_le_op_norm _ _ (by simp)
-
#align complex.im_clm_norm Complex.imClm_norm
@[simp]
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -28,30 +28,18 @@ open ContinuousLinearMap
namespace Complex
-/- warning: complex.det_conj_lie -> Complex.det_conjLie is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.det_conj_lie Complex.det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
det_conjAe
#align complex.det_conj_lie Complex.det_conjLie
-/- warning: complex.linear_equiv_det_conj_lie -> Complex.linearEquiv_det_conjLie is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
theorem linearEquiv_det_conjLie : conjLie.toLinearEquiv.det = -1 :=
linearEquiv_det_conjAe
#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLie
-/- warning: complex.re_clm_norm -> Complex.reClm_norm is a dubious translation:
-lean 3 declaration is
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.reClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
-but is expected to have type
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.reClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
-Case conversion may be inaccurate. Consider using '#align complex.re_clm_norm Complex.reClm_normₓ'. -/
@[simp]
theorem reClm_norm : ‖reClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -61,20 +49,11 @@ theorem reClm_norm : ‖reClm‖ = 1 :=
#align complex.re_clm_norm Complex.reClm_norm
-/- warning: complex.re_clm_nnnorm -> Complex.reClm_nnnorm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.re_clm_nnnorm Complex.reClm_nnnormₓ'. -/
@[simp]
theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
Subtype.ext reClm_norm
#align complex.re_clm_nnnorm Complex.reClm_nnnorm
-/- warning: complex.im_clm_norm -> Complex.imClm_norm is a dubious translation:
-lean 3 declaration is
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.imClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
-but is expected to have type
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.imClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
-Case conversion may be inaccurate. Consider using '#align complex.im_clm_norm Complex.imClm_normₓ'. -/
@[simp]
theorem imClm_norm : ‖imClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -84,44 +63,26 @@ theorem imClm_norm : ‖imClm‖ = 1 :=
#align complex.im_clm_norm Complex.imClm_norm
-/- warning: complex.im_clm_nnnorm -> Complex.imClm_nnnorm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.im_clm_nnnorm Complex.imClm_nnnormₓ'. -/
@[simp]
theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
Subtype.ext imClm_norm
#align complex.im_clm_nnnorm Complex.imClm_nnnorm
-/- warning: complex.conj_cle_norm -> Complex.conjCle_norm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.conj_cle_norm Complex.conjCle_normₓ'. -/
@[simp]
theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
conjLie.toLinearIsometry.norm_toContinuousLinearMap
#align complex.conj_cle_norm Complex.conjCle_norm
-/- warning: complex.conj_cle_nnorm -> Complex.conjCle_nnorm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.conj_cle_nnorm Complex.conjCle_nnormₓ'. -/
@[simp]
theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
Subtype.ext conjCle_norm
#align complex.conj_cle_nnorm Complex.conjCle_nnorm
-/- warning: complex.of_real_clm_norm -> Complex.ofRealClm_norm is a dubious translation:
-lean 3 declaration is
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real.module (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.ofRealClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
-but is expected to have type
- Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.ofRealClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
-Case conversion may be inaccurate. Consider using '#align complex.of_real_clm_norm Complex.ofRealClm_normₓ'. -/
@[simp]
theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
ofRealLi.norm_toContinuousLinearMap
#align complex.of_real_clm_norm Complex.ofRealClm_norm
-/- warning: complex.of_real_clm_nnnorm -> Complex.ofRealClm_nnnorm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnormₓ'. -/
@[simp]
theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
Subtype.ext <| ofRealClm_norm
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -29,10 +29,7 @@ open ContinuousLinearMap
namespace Complex
/- warning: complex.det_conj_lie -> Complex.det_conjLie is a dubious translation:
-lean 3 declaration is
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Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) (fun (_x : MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real 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-but is expected to have type
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+<too large>
Case conversion may be inaccurate. Consider using '#align complex.det_conj_lie Complex.det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
@@ -41,10 +38,7 @@ theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
#align complex.det_conj_lie Complex.det_conjLie
/- warning: complex.linear_equiv_det_conj_lie -> Complex.linearEquiv_det_conjLie is a dubious translation:
-lean 3 declaration is
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(RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) (Monoid.toMulOneClass.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (DivInvMonoid.toMonoid.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Group.toDivInvMonoid.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearEquiv.automorphismGroup.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module))))) (Units.mulOneClass.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) => (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) -> (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (MonoidHom.hasCoeToFun.{0, 0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) (Monoid.toMulOneClass.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (DivInvMonoid.toMonoid.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det._proof_1.{0} Real Real.commRing) (LinearEquiv.det._proof_2.{0} Real Real.commRing) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Group.toDivInvMonoid.{0} (LinearEquiv.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real 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(CommRing.toRing.{0} Real Real.commRing)))) (LinearEquiv.det.{0, 0} Real Real.commRing Complex Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module) Complex.conjLie)) (Neg.neg.{0} (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) (Units.hasNeg.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (NonUnitalNonAssocRing.toHasDistribNeg.{0} Real (NonAssocRing.toNonUnitalNonAssocRing.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring)))) (OfNat.ofNat.{0} (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) 1 (OfNat.mk.{0} (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) 1 (One.one.{0} (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) (MulOneClass.toHasOne.{0} (Units.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))) (Units.mulOneClass.{0} Real (Ring.toMonoid.{0} Real (CommRing.toRing.{0} Real Real.commRing))))))))
-but is expected to have type
- Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (FunLike.coe.{1, 1, 1} (MonoidHom.{0, 0} (LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (Units.{0} Real (MonoidWithZero.toMonoid.{0} 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Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivisionCommMonoid.toDivisionMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (CommGroup.toDivisionCommMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (Units.instCommGroupUnitsToMonoid.{0} Real Real.instCommMonoidReal)))))))))
+<too large>
Case conversion may be inaccurate. Consider using '#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
@@ -68,10 +62,7 @@ theorem reClm_norm : ‖reClm‖ = 1 :=
#align complex.re_clm_norm Complex.reClm_norm
/- warning: complex.re_clm_nnnorm -> Complex.reClm_nnnorm is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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(Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.reClm) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+<too large>
Case conversion may be inaccurate. Consider using '#align complex.re_clm_nnnorm Complex.reClm_nnnormₓ'. -/
@[simp]
theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
@@ -94,10 +85,7 @@ theorem imClm_norm : ‖imClm‖ = 1 :=
#align complex.im_clm_norm Complex.imClm_norm
/- warning: complex.im_clm_nnnorm -> Complex.imClm_nnnorm is a dubious translation:
-lean 3 declaration is
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+<too large>
Case conversion may be inaccurate. Consider using '#align complex.im_clm_nnnorm Complex.imClm_nnnormₓ'. -/
@[simp]
theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
@@ -105,10 +93,7 @@ theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
#align complex.im_clm_nnnorm Complex.imClm_nnnorm
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(One.toOfNat1.{0} Real Real.instOneReal))
+<too large>
Case conversion may be inaccurate. Consider using '#align complex.conj_cle_norm Complex.conjCle_normₓ'. -/
@[simp]
theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
@@ -116,10 +101,7 @@ theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
#align complex.conj_cle_norm Complex.conjCle_norm
/- warning: complex.conj_cle_nnorm -> Complex.conjCle_nnorm is a dubious translation:
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(NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField 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+<too large>
Case conversion may be inaccurate. Consider using '#align complex.conj_cle_nnorm Complex.conjCle_nnormₓ'. -/
@[simp]
theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
@@ -138,10 +120,7 @@ theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
#align complex.of_real_clm_norm Complex.ofRealClm_norm
/- warning: complex.of_real_clm_nnnorm -> Complex.ofRealClm_nnnorm is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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(NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) 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(NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.ofRealClm) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+<too large>
Case conversion may be inaccurate. Consider using '#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnormₓ'. -/
@[simp]
theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
! This file was ported from Lean 3 source module analysis.complex.operator_norm
-! leanprover-community/mathlib commit 468b141b14016d54b479eb7a0fff1e360b7e3cf6
+! leanprover-community/mathlib commit 33c67ae661dd8988516ff7f247b0be3018cdd952
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -14,6 +14,9 @@ import Mathbin.Data.Complex.Determinant
/-! # The basic continuous linear maps associated to `ℂ`
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
The continuous linear maps `complex.re_clm` (real part), `complex.im_clm` (imaginary part),
`complex.conj_cle` (conjugation), and `complex.of_real_clm` (inclusion of `ℝ`) were introduced in
`analysis.complex.operator_norm`. This file contains a few calculations requiring more imports:
mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4
@@ -29,7 +29,7 @@ namespace Complex
lean 3 declaration is
Eq.{1} Real (coeFn.{1, 1} (MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) (fun (_x : MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) => (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) -> Real) (MonoidHom.hasCoeToFun.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) (LinearMap.det.{0, 0} Complex Complex.addCommGroup Real Real.commRing (Complex.module.{0} Real Real.semiring Real.module)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (LinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (HasLiftT.mk.{1, 1} (LinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (CoeTCₓ.coe.{1, 1} (LinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (coeBase.{1, 1} (LinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (LinearEquiv.LinearMap.hasCoe.{0, 0, 0, 0} Real Real Complex Complex Real.semiring Real.semiring (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module) Complex.conjLie))) (Neg.neg.{0} Real Real.hasNeg (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))
but is expected to have type
- Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearMap.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} 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(NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} 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(Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex 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(NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie))) Real.instOneReal)))
+ Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearMap.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} 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Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) Real 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Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Real) (LinearEquiv.toLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))))) (AddCommGroup.toAddCommMonoid.{0} Complex (SeminormedAddCommGroup.toAddCommGroup.{0} Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex 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(NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie))) Real.instOneReal)))
Case conversion may be inaccurate. Consider using '#align complex.det_conj_lie Complex.det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
@@ -41,7 +41,7 @@ theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
lean 3 declaration is
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Real.commRing))))))))
but is expected to have type
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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivInvOneMonoid.toInvOneClass.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivisionMonoid.toDivInvOneMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivisionCommMonoid.toDivisionMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (CommGroup.toDivisionCommMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (Units.instCommGroupUnitsToMonoid.{0} Real Real.instCommMonoidReal)))))))))
+ Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (FunLike.coe.{1, 1, 1} (MonoidHom.{0, 0} (LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) 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(x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivisionMonoid.toDivInvOneMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivisionCommMonoid.toDivisionMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (CommGroup.toDivisionCommMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (Units.instCommGroupUnitsToMonoid.{0} Real Real.instCommMonoidReal)))))))))
Case conversion may be inaccurate. Consider using '#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e
@@ -25,18 +25,36 @@ open ContinuousLinearMap
namespace Complex
+/- warning: complex.det_conj_lie -> Complex.det_conjLie is a dubious translation:
+lean 3 declaration is
+ Eq.{1} Real (coeFn.{1, 1} (MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) (fun (_x : MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocRing.toNonAssocSemiring.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Ring.toNonAssocRing.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) => (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) -> Real) (MonoidHom.hasCoeToFun.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) Real (MulZeroOneClass.toMulOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (NonAssocSemiring.toMulZeroOneClass.{0} (LinearMap.{0, 0, 0, 0} Real Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring 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Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (Module.End.ring.{0, 0} Real Complex (Ring.toSemiring.{0} Real (CommRing.toRing.{0} Real Real.commRing)) Complex.addCommGroup (Complex.module.{0} Real Real.semiring Real.module)))))) (MulZeroOneClass.toMulOneClass.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real (CommRing.toRing.{0} Real Real.commRing)))))) (LinearMap.det.{0, 0} Complex Complex.addCommGroup Real Real.commRing (Complex.module.{0} Real Real.semiring Real.module)) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (LinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) 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Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module) Complex.conjLie))) (Neg.neg.{0} Real Real.hasNeg (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))
+but is expected to have type
+ Eq.{1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearMap.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} 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(Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie))) (FunLike.coe.{1, 1, 1} (MonoidHom.{0, 0} (LinearMap.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) Complex Complex (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (AddCommGroup.toAddCommMonoid.{0} 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+Case conversion may be inaccurate. Consider using '#align complex.det_conj_lie Complex.det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear map. -/
@[simp]
theorem det_conjLie : (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ).det = -1 :=
det_conjAe
#align complex.det_conj_lie Complex.det_conjLie
+/- warning: complex.linear_equiv_det_conj_lie -> Complex.linearEquiv_det_conjLie is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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(Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex 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(NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (DivInvOneMonoid.toInvOneClass.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real 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(NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) => Units.{0} Real (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing))))) (LinearIsometryEquiv.toLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex 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(NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (CommGroup.toDivisionCommMonoid.{0} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : LinearEquiv.{0, 0, 0, 0} Real Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real (CommSemiring.toSemiring.{0} Real (CommRing.toCommSemiring.{0} Real Real.commRing)))) (RingHomInvPair.ids.{0} Real (CommSemiring.toSemiring.{0} Real 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(NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjLie)) (Units.instCommGroupUnitsToMonoid.{0} Real Real.instCommMonoidReal)))))))))
+Case conversion may be inaccurate. Consider using '#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLieₓ'. -/
/-- The determinant of `conj_lie`, as a linear equiv. -/
@[simp]
theorem linearEquiv_det_conjLie : conjLie.toLinearEquiv.det = -1 :=
linearEquiv_det_conjAe
#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLie
+/- warning: complex.re_clm_norm -> Complex.reClm_norm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.reClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.reClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align complex.re_clm_norm Complex.reClm_normₓ'. -/
@[simp]
theorem reClm_norm : ‖reClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -46,11 +64,23 @@ theorem reClm_norm : ‖reClm‖ = 1 :=
#align complex.re_clm_norm Complex.reClm_norm
+/- warning: complex.re_clm_nnnorm -> Complex.reClm_nnnorm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.reClm) (OfNat.ofNat.{0} NNReal 1 (OfNat.mk.{0} NNReal 1 (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
+but is expected to have type
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.reClm) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+Case conversion may be inaccurate. Consider using '#align complex.re_clm_nnnorm Complex.reClm_nnnormₓ'. -/
@[simp]
theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
Subtype.ext reClm_norm
#align complex.re_clm_nnnorm Complex.reClm_nnnorm
+/- warning: complex.im_clm_norm -> Complex.imClm_norm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.imClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.imClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align complex.im_clm_norm Complex.imClm_normₓ'. -/
@[simp]
theorem imClm_norm : ‖imClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
@@ -60,26 +90,56 @@ theorem imClm_norm : ‖imClm‖ = 1 :=
#align complex.im_clm_norm Complex.imClm_norm
+/- warning: complex.im_clm_nnnorm -> Complex.imClm_nnnorm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid (Complex.module.{0} Real Real.semiring Real.module) Real.module) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.imClm) (OfNat.ofNat.{0} NNReal 1 (OfNat.mk.{0} NNReal 1 (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
+but is expected to have type
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Real (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.imClm) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+Case conversion may be inaccurate. Consider using '#align complex.im_clm_nnnorm Complex.imClm_nnnormₓ'. -/
@[simp]
theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
Subtype.ext imClm_norm
#align complex.im_clm_nnnorm Complex.imClm_nnnorm
+/- warning: complex.conj_cle_norm -> Complex.conjCle_norm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (HasLiftT.mk.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (CoeTCₓ.coe.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (coeBase.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module))))) Complex.conjCle)) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) (ContinuousLinearEquiv.toContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjCle)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align complex.conj_cle_norm Complex.conjCle_normₓ'. -/
@[simp]
theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
conjLie.toLinearIsometry.norm_toContinuousLinearMap
#align complex.conj_cle_norm Complex.conjCle_norm
+/- warning: complex.conj_cle_nnorm -> Complex.conjCle_nnorm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Complex Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (HasLiftT.mk.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (CoeTCₓ.coe.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (coeBase.{1, 1} (ContinuousLinearEquiv.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearEquiv.ContinuousLinearMap.coe.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) (Complex.module.{0} Real Real.semiring Real.module) (Complex.module.{0} Real Real.semiring Real.module))))) Complex.conjCle)) (OfNat.ofNat.{0} NNReal 1 (OfNat.mk.{0} NNReal 1 (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
+but is expected to have type
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex 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(NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex 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(NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) (ContinuousLinearEquiv.toContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomInvPair.ids.{0} Real Real.semiring) (RingHomInvPair.ids.{0} Real Real.semiring) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField))) Complex.conjCle)) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+Case conversion may be inaccurate. Consider using '#align complex.conj_cle_nnorm Complex.conjCle_nnormₓ'. -/
@[simp]
theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
Subtype.ext conjCle_norm
#align complex.conj_cle_nnorm Complex.conjCle_nnorm
+/- warning: complex.of_real_clm_norm -> Complex.ofRealClm_norm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real.module (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.ofRealClm) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))
+but is expected to have type
+ Eq.{1} Real (Norm.norm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (ContinuousLinearMap.hasOpNorm.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring))) Complex.ofRealClm) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))
+Case conversion may be inaccurate. Consider using '#align complex.of_real_clm_norm Complex.ofRealClm_normₓ'. -/
@[simp]
theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
ofRealLi.norm_toContinuousLinearMap
#align complex.of_real_clm_norm Complex.ofRealClm_norm
+/- warning: complex.of_real_clm_nnnorm -> Complex.ofRealClm_nnnorm is a dubious translation:
+lean 3 declaration is
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real.module (Complex.module.{0} Real Real.semiring Real.module)) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real.module (Complex.module.{0} Real Real.semiring Real.module)) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.addCommMonoid Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSemiNormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField)))))) (AddCommGroup.toAddCommMonoid.{0} Complex Complex.addCommGroup) Real.module (Complex.module.{0} Real Real.semiring Real.module)) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.normedField))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.normedAddCommGroup (NormedField.toNormedSpace.{0} Complex Complex.normedField)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.ofRealClm) (OfNat.ofNat.{0} NNReal 1 (OfNat.mk.{0} NNReal 1 (One.one.{0} NNReal (AddMonoidWithOne.toOne.{0} NNReal (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNReal (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNReal (Semiring.toNonAssocSemiring.{0} NNReal NNReal.semiring)))))))
+but is expected to have type
+ Eq.{1} NNReal (NNNorm.nnnorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (SeminormedAddGroup.toNNNorm.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (SeminormedAddCommGroup.toSeminormedAddGroup.{0} (ContinuousLinearMap.{0, 0, 0, 0} Real Real Real.semiring Real.semiring (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real.instAddCommMonoidReal Complex (UniformSpace.toTopologicalSpace.{0} Complex (PseudoMetricSpace.toUniformSpace.{0} Complex (SeminormedRing.toPseudoMetricSpace.{0} Complex (SeminormedCommRing.toSeminormedRing.{0} Complex (NormedCommRing.toSeminormedCommRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Complex (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Complex (NonAssocRing.toNonUnitalNonAssocRing.{0} Complex (Ring.toNonAssocRing.{0} Complex Complex.instRingComplex)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)) (Complex.instModuleComplexToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingInstRingComplex.{0} Real Real.semiring (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NormedField.toNormedSpace.{0} Real Real.normedField)))) (ContinuousLinearMap.toSeminormedAddCommGroup.{0, 0, 0, 0} Real Real Real Complex (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Real (NormedRing.toNonUnitalNormedRing.{0} Real (NormedCommRing.toNormedRing.{0} Real Real.normedCommRing)))) (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Complex (NonUnitalNormedRing.toNonUnitalSeminormedRing.{0} Complex (NormedRing.toNonUnitalNormedRing.{0} Complex (NormedCommRing.toNormedRing.{0} Complex (NormedField.toNormedCommRing.{0} Complex Complex.instNormedFieldComplex))))) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) (NormedField.toNormedSpace.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField))) (NormedSpace.complexToReal.{0} Complex Complex.instNormedAddCommGroupComplex (NormedField.toNormedSpace.{0} Complex Complex.instNormedFieldComplex)) (RingHom.id.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) (RingHomIsometric.ids.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) Complex.ofRealClm) (OfNat.ofNat.{0} NNReal 1 (One.toOfNat1.{0} NNReal instNNRealOne))
+Case conversion may be inaccurate. Consider using '#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnormₓ'. -/
@[simp]
theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
Subtype.ext <| ofRealClm_norm
mathlib commit https://github.com/leanprover-community/mathlib/commit/ddec54a71a0dd025c05445d467f1a2b7d586a3ba
@@ -55,7 +55,7 @@ theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
theorem imClm_norm : ‖imClm‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
- 1 = ‖imClm i‖ := by simp
+ 1 = ‖imClm I‖ := by simp
_ ≤ ‖imClm‖ := unit_le_op_norm _ _ (by simp)
#align complex.im_clm_norm Complex.imClm_norm
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Split the 2300-line behemoth OperatorNorm.lean
into 8 smaller files, of which the largest is 600 lines.
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.NormedSpace.OperatorNorm
+import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
@@ -17,7 +17,6 @@ The continuous linear maps `Complex.reCLM` (real part), `Complex.imCLM` (imagina
the operator norm and (for `Complex.conjCLE`) the determinant.
-/
-
open ContinuousLinearMap
namespace Complex
@@ -39,7 +39,7 @@ theorem reCLM_norm : ‖reCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖reCLM 1‖ := by simp
- _ ≤ ‖reCLM‖ := unit_le_op_norm _ _ (by simp)
+ _ ≤ ‖reCLM‖ := unit_le_opNorm _ _ (by simp)
#align complex.re_clm_norm Complex.reCLM_norm
@[simp]
@@ -52,7 +52,7 @@ theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖imCLM I‖ := by simp
- _ ≤ ‖imCLM‖ := unit_le_op_norm _ _ (by simp)
+ _ ≤ ‖imCLM‖ := unit_le_opNorm _ _ (by simp)
#align complex.im_clm_norm Complex.imCLM_norm
@[simp]
Clm
-> CLM
, Cle
-> CLE
(#10018)
Rename
Complex.equivRealProdClm
→ Complex.equivRealProdCLM
;
CLE
?Complex.reClm
→ Complex.reCLM
;Complex.imClm
→ Complex.imCLM
;Complex.conjLie
→ Complex.conjLIE
;Complex.conjCle
→ Complex.conjCLE
;Complex.ofRealLi
→ Complex.ofRealLI
;Complex.ofRealClm
→ Complex.ofRealCLM
;fderivInnerClm
→ fderivInnerCLM
;LinearPMap.adjointDomainMkClm
→ LinearPMap.adjointDomainMkCLM
;LinearPMap.adjointDomainMkClmExtend
→ LinearPMap.adjointDomainMkCLMExtend
;IsROrC.reClm
→ IsROrC.reCLM
;IsROrC.imClm
→ IsROrC.imCLM
;IsROrC.conjLie
→ IsROrC.conjLIE
;IsROrC.conjCle
→ IsROrC.conjCLE
;IsROrC.ofRealLi
→ IsROrC.ofRealLI
;IsROrC.ofRealClm
→ IsROrC.ofRealCLM
;MeasureTheory.condexpL1Clm
→ MeasureTheory.condexpL1CLM
;algebraMapClm
→ algebraMapCLM
;WeakDual.CharacterSpace.toClm
→ WeakDual.CharacterSpace.toCLM
;BoundedContinuousFunction.evalClm
→ BoundedContinuousFunction.evalCLM
;ContinuousMap.evalClm
→ ContinuousMap.evalCLM
;TrivSqZeroExt.fstClm
→ TrivSqZeroExt.fstClm
;TrivSqZeroExt.sndClm
→ TrivSqZeroExt.sndCLM
;TrivSqZeroExt.inlClm
→ TrivSqZeroExt.inlCLM
;TrivSqZeroExt.inrClm
→ TrivSqZeroExt.inrCLM
and related theorems.
@@ -11,10 +11,10 @@ import Mathlib.Data.Complex.Determinant
/-! # The basic continuous linear maps associated to `ℂ`
-The continuous linear maps `Complex.reClm` (real part), `Complex.imClm` (imaginary part),
-`Complex.conjCle` (conjugation), and `Complex.ofRealClm` (inclusion of `ℝ`) were introduced in
+The continuous linear maps `Complex.reCLM` (real part), `Complex.imCLM` (imaginary part),
+`Complex.conjCLE` (conjugation), and `Complex.ofRealCLM` (inclusion of `ℝ`) were introduced in
`Analysis.Complex.Basic`. This file contains a few calculations requiring more imports:
-the operator norm and (for `Complex.conjCle`) the determinant.
+the operator norm and (for `Complex.conjCLE`) the determinant.
-/
@@ -22,62 +22,62 @@ open ContinuousLinearMap
namespace Complex
-/-- The determinant of `conjLie`, as a linear map. -/
+/-- The determinant of `conjLIE`, as a linear map. -/
@[simp]
-theorem det_conjLie : LinearMap.det (conjLie.toLinearEquiv : ℂ →ₗ[ℝ] ℂ) = -1 :=
+theorem det_conjLIE : LinearMap.det (conjLIE.toLinearEquiv : ℂ →ₗ[ℝ] ℂ) = -1 :=
det_conjAe
-#align complex.det_conj_lie Complex.det_conjLie
+#align complex.det_conj_lie Complex.det_conjLIE
-/-- The determinant of `conjLie`, as a linear equiv. -/
+/-- The determinant of `conjLIE`, as a linear equiv. -/
@[simp]
-theorem linearEquiv_det_conjLie : LinearEquiv.det conjLie.toLinearEquiv = -1 :=
+theorem linearEquiv_det_conjLIE : LinearEquiv.det conjLIE.toLinearEquiv = -1 :=
linearEquiv_det_conjAe
-#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLie
+#align complex.linear_equiv_det_conj_lie Complex.linearEquiv_det_conjLIE
@[simp]
-theorem reClm_norm : ‖reClm‖ = 1 :=
+theorem reCLM_norm : ‖reCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
- 1 = ‖reClm 1‖ := by simp
- _ ≤ ‖reClm‖ := unit_le_op_norm _ _ (by simp)
-#align complex.re_clm_norm Complex.reClm_norm
+ 1 = ‖reCLM 1‖ := by simp
+ _ ≤ ‖reCLM‖ := unit_le_op_norm _ _ (by simp)
+#align complex.re_clm_norm Complex.reCLM_norm
@[simp]
-theorem reClm_nnnorm : ‖reClm‖₊ = 1 :=
- Subtype.ext reClm_norm
-#align complex.re_clm_nnnorm Complex.reClm_nnnorm
+theorem reCLM_nnnorm : ‖reCLM‖₊ = 1 :=
+ Subtype.ext reCLM_norm
+#align complex.re_clm_nnnorm Complex.reCLM_nnnorm
@[simp]
-theorem imClm_norm : ‖imClm‖ = 1 :=
+theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
- 1 = ‖imClm I‖ := by simp
- _ ≤ ‖imClm‖ := unit_le_op_norm _ _ (by simp)
-#align complex.im_clm_norm Complex.imClm_norm
+ 1 = ‖imCLM I‖ := by simp
+ _ ≤ ‖imCLM‖ := unit_le_op_norm _ _ (by simp)
+#align complex.im_clm_norm Complex.imCLM_norm
@[simp]
-theorem imClm_nnnorm : ‖imClm‖₊ = 1 :=
- Subtype.ext imClm_norm
-#align complex.im_clm_nnnorm Complex.imClm_nnnorm
+theorem imCLM_nnnorm : ‖imCLM‖₊ = 1 :=
+ Subtype.ext imCLM_norm
+#align complex.im_clm_nnnorm Complex.imCLM_nnnorm
@[simp]
-theorem conjCle_norm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖ = 1 :=
- conjLie.toLinearIsometry.norm_toContinuousLinearMap
-#align complex.conj_cle_norm Complex.conjCle_norm
+theorem conjCLE_norm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖ = 1 :=
+ conjLIE.toLinearIsometry.norm_toContinuousLinearMap
+#align complex.conj_cle_norm Complex.conjCLE_norm
@[simp]
-theorem conjCle_nnorm : ‖(conjCle : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
- Subtype.ext conjCle_norm
-#align complex.conj_cle_nnorm Complex.conjCle_nnorm
+theorem conjCLE_nnorm : ‖(conjCLE : ℂ →L[ℝ] ℂ)‖₊ = 1 :=
+ Subtype.ext conjCLE_norm
+#align complex.conj_cle_nnorm Complex.conjCLE_nnorm
@[simp]
-theorem ofRealClm_norm : ‖ofRealClm‖ = 1 :=
- ofRealLi.norm_toContinuousLinearMap
-#align complex.of_real_clm_norm Complex.ofRealClm_norm
+theorem ofRealCLM_norm : ‖ofRealCLM‖ = 1 :=
+ ofRealLI.norm_toContinuousLinearMap
+#align complex.of_real_clm_norm Complex.ofRealCLM_norm
@[simp]
-theorem ofRealClm_nnnorm : ‖ofRealClm‖₊ = 1 :=
- Subtype.ext <| ofRealClm_norm
-#align complex.of_real_clm_nnnorm Complex.ofRealClm_nnnorm
+theorem ofRealCLM_nnnorm : ‖ofRealCLM‖₊ = 1 :=
+ Subtype.ext <| ofRealCLM_norm
+#align complex.of_real_clm_nnnorm Complex.ofRealCLM_nnnorm
end Complex
@@ -2,16 +2,13 @@
Copyright (c) Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module analysis.complex.operator_norm
-! leanprover-community/mathlib commit 468b141b14016d54b479eb7a0fff1e360b7e3cf6
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm
import Mathlib.Data.Complex.Determinant
+#align_import analysis.complex.operator_norm from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
+
/-! # The basic continuous linear maps associated to `ℂ`
The continuous linear maps `Complex.reClm` (real part), `Complex.imClm` (imaginary part),
The unported dependencies are
algebra.order.module
init.core
linear_algebra.free_module.finite.rank
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
linear_algebra.free_module.rank
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file