deprecated.subring
⟷
Mathlib.Deprecated.Subring
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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(last sync)
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -163,7 +163,7 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
have h0 : C 0 := add_neg_self (1 : R) ▸ ha h1 hneg1
rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩; clear hx
induction' L with hd tl ih; · exact h0
- rw [List.forall_mem_cons] at HL
+ rw [List.forall_mem_cons] at HL
suffices C (List.prod hd) by
rw [List.map_cons, List.sum_cons]
exact ha this (ih HL.2)
@@ -171,16 +171,16 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
rsuffices ⟨L, HL', HP | HP⟩ :
∃ L : List R, (∀ x ∈ L, x ∈ s) ∧ (List.prod hd = List.prod L ∨ List.prod hd = -List.prod L)
· rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact h1
- rw [List.forall_mem_cons] at HL'
+ rw [List.forall_mem_cons] at HL'
rw [List.prod_cons]
exact hs _ HL'.1 _ (ih HL'.2)
· rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact hneg1
rw [List.prod_cons, neg_mul_eq_mul_neg]
- rw [List.forall_mem_cons] at HL'
+ rw [List.forall_mem_cons] at HL'
exact hs _ HL'.1 _ (ih HL'.2)
induction' hd with hd tl ih
· exact ⟨[], List.forall_mem_nil _, Or.inl rfl⟩
- rw [List.forall_mem_cons] at HL
+ rw [List.forall_mem_cons] at HL
rcases ih HL.2 with ⟨L, HL', HP | HP⟩ <;> cases' HL.1 with hhd hhd
·
exact
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,9 +3,9 @@ Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
-import Mathbin.Deprecated.Subgroup
-import Mathbin.Deprecated.Group
-import Mathbin.RingTheory.Subring.Basic
+import Deprecated.Subgroup
+import Deprecated.Group
+import RingTheory.Subring.Basic
#align_import deprecated.subring from "leanprover-community/mathlib"@"23aa88e32dcc9d2a24cca7bc23268567ed4cd7d6"
mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504
@@ -51,7 +51,7 @@ def IsSubring.subring {S : Set R} (hs : IsSubring S) : Subring R
where
carrier := S
one_mem' := hs.one_mem
- mul_mem' _ _ := hs.mul_mem
+ hMul_mem' _ _ := hs.hMul_mem
zero_mem' := hs.zero_mem
add_mem' _ _ := hs.add_mem
neg_mem' _ := hs.neg_mem
@@ -201,11 +201,11 @@ theorem closure.isSubring : IsSubring (closure s) :=
AddGroup.closure.isAddSubgroup
_ with
one_mem := AddGroup.mem_closure <| IsSubmonoid.one_mem <| Monoid.closure.isSubmonoid _
- mul_mem := fun a b ha hb =>
+ hMul_mem := fun a b ha hb =>
AddGroup.InClosure.rec_on hb
(fun c hc =>
AddGroup.InClosure.rec_on ha
- (fun d hd => AddGroup.subset_closure ((Monoid.closure.isSubmonoid _).mul_mem hd hc))
+ (fun d hd => AddGroup.subset_closure ((Monoid.closure.isSubmonoid _).hMul_mem hd hc))
((MulZeroClass.zero_mul c).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
(fun d hd hdc =>
neg_mul_eq_neg_mul d c ▸ (AddGroup.closure.isAddSubgroup _).neg_mem hdc)
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,16 +2,13 @@
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module deprecated.subring
-! leanprover-community/mathlib commit 23aa88e32dcc9d2a24cca7bc23268567ed4cd7d6
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Deprecated.Subgroup
import Mathbin.Deprecated.Group
import Mathbin.RingTheory.Subring.Basic
+#align_import deprecated.subring from "leanprover-community/mathlib"@"23aa88e32dcc9d2a24cca7bc23268567ed4cd7d6"
+
/-!
# Unbundled subrings (deprecated)
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -63,45 +63,57 @@ def IsSubring.subring {S : Set R} (hs : IsSubring S) : Subring R
namespace RingHom
+#print RingHom.isSubring_preimage /-
theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
{ IsAddGroupHom.preimage f.to_isAddGroupHom hs.to_isAddSubgroup,
IsSubmonoid.preimage f.to_isMonoidHom hs.to_isSubmonoid with }
#align ring_hom.is_subring_preimage RingHom.isSubring_preimage
+-/
+#print RingHom.isSubring_image /-
theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
{ IsAddGroupHom.image_addSubgroup f.to_isAddGroupHom hs.to_isAddSubgroup,
IsSubmonoid.image f.to_isMonoidHom hs.to_isSubmonoid with }
#align ring_hom.is_subring_image RingHom.isSubring_image
+-/
+#print RingHom.isSubring_set_range /-
theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
{ IsAddGroupHom.range_addSubgroup f.to_isAddGroupHom, Range.isSubmonoid f.to_isMonoidHom with }
#align ring_hom.is_subring_set_range RingHom.isSubring_set_range
+-/
end RingHom
variable {cR : Type u} [CommRing cR]
+#print IsSubring.inter /-
theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : IsSubring S₂) :
IsSubring (S₁ ∩ S₂) :=
{ IsAddSubgroup.inter hS₁.to_isAddSubgroup hS₂.to_isAddSubgroup,
IsSubmonoid.inter hS₁.to_isSubmonoid hS₂.to_isSubmonoid with }
#align is_subring.inter IsSubring.inter
+-/
+#print IsSubring.iInter /-
theorem IsSubring.iInter {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
IsSubring (Set.iInter S) :=
{ IsAddSubgroup.iInter fun i => (h i).to_isAddSubgroup,
IsSubmonoid.iInter fun i => (h i).to_isSubmonoid with }
#align is_subring.Inter IsSubring.iInter
+-/
+#print isSubring_iUnion_of_directed /-
theorem isSubring_iUnion_of_directed {ι : Type _} [hι : Nonempty ι] {s : ι → Set R}
(h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubring (⋃ i, s i) :=
{ to_isAddSubgroup := isAddSubgroup_iUnion_of_directed (fun i => (h i).to_isAddSubgroup) Directed
to_isSubmonoid := isSubmonoid_iUnion_of_directed (fun i => (h i).to_isSubmonoid) Directed }
#align is_subring_Union_of_directed isSubring_iUnion_of_directed
+-/
namespace Ring
@@ -117,6 +129,7 @@ variable {s : Set R}
attribute [local reducible] closure
+#print Ring.exists_list_of_mem_closure /-
theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
∃ L : List (List R), (∀ l ∈ L, ∀ x ∈ l, x ∈ s ∨ x = (-1 : R)) ∧ (L.map List.prod).Sum = a :=
AddGroup.InClosure.rec_on h
@@ -141,9 +154,11 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
| _, _, ⟨L1, h1, rfl⟩, ⟨L2, h2, rfl⟩ =>
⟨L1 ++ L2, List.forall_mem_append.2 ⟨h1, h2⟩, by rw [List.map_append, List.sum_append]⟩
#align ring.exists_list_of_mem_closure Ring.exists_list_of_mem_closure
+-/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print Ring.InClosure.recOn /-
@[elab_as_elim]
protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s) (h1 : C 1)
(hneg1 : C (-1)) (hs : ∀ z ∈ s, ∀ n, C n → C (z * n)) (ha : ∀ {x y}, C x → C y → C (x + y)) :
@@ -181,6 +196,7 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
Or.inr <| by rw [List.prod_cons, List.prod_cons, HP, neg_mul_eq_mul_neg]⟩
· exact ⟨L, HL', Or.inl <| by rw [List.prod_cons, hhd, HP, neg_one_mul, neg_neg]⟩
#align ring.in_closure.rec_on Ring.InClosure.recOn
+-/
#print Ring.closure.isSubring /-
theorem closure.isSubring : IsSubring (closure s) :=
@@ -235,6 +251,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
#align ring.closure_mono Ring.closure_mono
-/
+#print Ring.image_closure /-
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
le_antisymm
@@ -252,6 +269,7 @@ theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
(closure_subset (RingHom.isSubring_image _ closure.isSubring) <|
Set.image_subset _ subset_closure)
#align ring.image_closure Ring.image_closure
+-/
end Ring
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -151,7 +151,7 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
have h0 : C 0 := add_neg_self (1 : R) ▸ ha h1 hneg1
rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩; clear hx
induction' L with hd tl ih; · exact h0
- rw [List.forall_mem_cons] at HL
+ rw [List.forall_mem_cons] at HL
suffices C (List.prod hd) by
rw [List.map_cons, List.sum_cons]
exact ha this (ih HL.2)
@@ -159,16 +159,16 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
rsuffices ⟨L, HL', HP | HP⟩ :
∃ L : List R, (∀ x ∈ L, x ∈ s) ∧ (List.prod hd = List.prod L ∨ List.prod hd = -List.prod L)
· rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact h1
- rw [List.forall_mem_cons] at HL'
+ rw [List.forall_mem_cons] at HL'
rw [List.prod_cons]
exact hs _ HL'.1 _ (ih HL'.2)
· rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact hneg1
rw [List.prod_cons, neg_mul_eq_mul_neg]
- rw [List.forall_mem_cons] at HL'
+ rw [List.forall_mem_cons] at HL'
exact hs _ HL'.1 _ (ih HL'.2)
induction' hd with hd tl ih
· exact ⟨[], List.forall_mem_nil _, Or.inl rfl⟩
- rw [List.forall_mem_cons] at HL
+ rw [List.forall_mem_cons] at HL
rcases ih HL.2 with ⟨L, HL', HP | HP⟩ <;> cases' HL.1 with hhd hhd
·
exact
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -63,36 +63,18 @@ def IsSubring.subring {S : Set R} (hs : IsSubring S) : Subring R
namespace RingHom
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theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
{ IsAddGroupHom.preimage f.to_isAddGroupHom hs.to_isAddSubgroup,
IsSubmonoid.preimage f.to_isMonoidHom hs.to_isSubmonoid with }
#align ring_hom.is_subring_preimage RingHom.isSubring_preimage
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theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
{ IsAddGroupHom.image_addSubgroup f.to_isAddGroupHom hs.to_isAddSubgroup,
IsSubmonoid.image f.to_isMonoidHom hs.to_isSubmonoid with }
#align ring_hom.is_subring_image RingHom.isSubring_image
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theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
{ IsAddGroupHom.range_addSubgroup f.to_isAddGroupHom, Range.isSubmonoid f.to_isMonoidHom with }
@@ -102,36 +84,18 @@ end RingHom
variable {cR : Type u} [CommRing cR]
-/- warning: is_subring.inter -> IsSubring.inter is a dubious translation:
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- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {S₁ : Set.{u1} R} {S₂ : Set.{u1} R}, (IsSubring.{u1} R _inst_1 S₁) -> (IsSubring.{u1} R _inst_1 S₂) -> (IsSubring.{u1} R _inst_1 (Inter.inter.{u1} (Set.{u1} R) (Set.hasInter.{u1} R) S₁ S₂))
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-Case conversion may be inaccurate. Consider using '#align is_subring.inter IsSubring.interₓ'. -/
theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : IsSubring S₂) :
IsSubring (S₁ ∩ S₂) :=
{ IsAddSubgroup.inter hS₁.to_isAddSubgroup hS₂.to_isAddSubgroup,
IsSubmonoid.inter hS₁.to_isSubmonoid hS₂.to_isSubmonoid with }
#align is_subring.inter IsSubring.inter
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-Case conversion may be inaccurate. Consider using '#align is_subring.Inter IsSubring.iInterₓ'. -/
theorem IsSubring.iInter {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
IsSubring (Set.iInter S) :=
{ IsAddSubgroup.iInter fun i => (h i).to_isAddSubgroup,
IsSubmonoid.iInter fun i => (h i).to_isSubmonoid with }
#align is_subring.Inter IsSubring.iInter
-/- warning: is_subring_Union_of_directed -> isSubring_iUnion_of_directed is a dubious translation:
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- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {ι : Type.{u1}} [hι : Nonempty.{succ u1} ι] {s : ι -> (Set.{u2} R)}, (forall (i : ι), IsSubring.{u2} R _inst_1 (s i)) -> (forall (i : ι) (j : ι), Exists.{succ u1} ι (fun (k : ι) => And (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s i) (s k)) (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s j) (s k)))) -> (IsSubring.{u2} R _inst_1 (Set.iUnion.{u2, succ u1} R ι (fun (i : ι) => s i)))
-Case conversion may be inaccurate. Consider using '#align is_subring_Union_of_directed isSubring_iUnion_of_directedₓ'. -/
theorem isSubring_iUnion_of_directed {ι : Type _} [hι : Nonempty ι] {s : ι → Set R}
(h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubring (⋃ i, s i) :=
@@ -153,12 +117,6 @@ variable {s : Set R}
attribute [local reducible] closure
-/- warning: ring.exists_list_of_mem_closure -> Ring.exists_list_of_mem_closure is a dubious translation:
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- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.instMembershipList.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.mem.{u1, u1} R (List.{u1} R) (List.instMembershipList.{u1} R) x l) -> (Or (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))) L)) a)))
-Case conversion may be inaccurate. Consider using '#align ring.exists_list_of_mem_closure Ring.exists_list_of_mem_closureₓ'. -/
theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
∃ L : List (List R), (∀ l ∈ L, ∀ x ∈ l, x ∈ s ∨ x = (-1 : R)) ∧ (L.map List.prod).Sum = a :=
AddGroup.InClosure.rec_on h
@@ -184,12 +142,6 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
⟨L1 ++ L2, List.forall_mem_append.2 ⟨h1, h2⟩, by rw [List.map_append, List.sum_append]⟩
#align ring.exists_list_of_mem_closure Ring.exists_list_of_mem_closure
-/- warning: ring.in_closure.rec_on -> Ring.InClosure.recOn is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align ring.in_closure.rec_on Ring.InClosure.recOnₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@[elab_as_elim]
@@ -283,12 +235,6 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
#align ring.closure_mono Ring.closure_mono
-/
-/- warning: ring.image_closure -> Ring.image_closure is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align ring.image_closure Ring.image_closureₓ'. -/
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
le_antisymm
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -197,29 +197,20 @@ protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s
(hneg1 : C (-1)) (hs : ∀ z ∈ s, ∀ n, C n → C (z * n)) (ha : ∀ {x y}, C x → C y → C (x + y)) :
C x := by
have h0 : C 0 := add_neg_self (1 : R) ▸ ha h1 hneg1
- rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩
- clear hx
- induction' L with hd tl ih
- · exact h0
+ rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩; clear hx
+ induction' L with hd tl ih; · exact h0
rw [List.forall_mem_cons] at HL
suffices C (List.prod hd) by
rw [List.map_cons, List.sum_cons]
exact ha this (ih HL.2)
- replace HL := HL.1
- clear ih tl
+ replace HL := HL.1; clear ih tl
rsuffices ⟨L, HL', HP | HP⟩ :
∃ L : List R, (∀ x ∈ L, x ∈ s) ∧ (List.prod hd = List.prod L ∨ List.prod hd = -List.prod L)
- · rw [HP]
- clear HP HL hd
- induction' L with hd tl ih
- · exact h1
+ · rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact h1
rw [List.forall_mem_cons] at HL'
rw [List.prod_cons]
exact hs _ HL'.1 _ (ih HL'.2)
- · rw [HP]
- clear HP HL hd
- induction' L with hd tl ih
- · exact hneg1
+ · rw [HP]; clear HP HL hd; induction' L with hd tl ih; · exact hneg1
rw [List.prod_cons, neg_mul_eq_mul_neg]
rw [List.forall_mem_cons] at HL'
exact hs _ HL'.1 _ (ih HL'.2)
@@ -304,17 +295,14 @@ theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
(by
rintro _ ⟨x, hx, rfl⟩
apply in_closure.rec_on hx <;> intros
- · rw [f.map_one]
- apply closure.is_subring.to_is_submonoid.one_mem
+ · rw [f.map_one]; apply closure.is_subring.to_is_submonoid.one_mem
· rw [f.map_neg, f.map_one]
apply closure.is_subring.to_is_add_subgroup.neg_mem
apply closure.is_subring.to_is_submonoid.one_mem
· rw [f.map_mul]
apply closure.is_subring.to_is_submonoid.mul_mem <;>
solve_by_elim [subset_closure, Set.mem_image_of_mem]
- · rw [f.map_add]
- apply closure.is_subring.to_is_add_submonoid.add_mem
- assumption')
+ · rw [f.map_add]; apply closure.is_subring.to_is_add_submonoid.add_mem; assumption')
(closure_subset (RingHom.isSubring_image _ closure.isSubring) <|
Set.image_subset _ subset_closure)
#align ring.image_closure Ring.image_closure
mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4
@@ -67,7 +67,7 @@ namespace RingHom
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_preimage RingHom.isSubring_preimageₓ'. -/
theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
@@ -79,7 +79,7 @@ theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_image RingHom.isSubring_imageₓ'. -/
theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
@@ -91,7 +91,7 @@ theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+*
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_set_range RingHom.isSubring_set_rangeₓ'. -/
theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
@@ -296,7 +296,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {S : Type.{u2}} [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (s : Set.{u1} R), Eq.{succ u2} (Set.{u2} S) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) (Ring.closure.{u1} R _inst_1 s)) (Ring.closure.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) s))
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring.image_closure Ring.image_closureₓ'. -/
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -114,30 +114,30 @@ theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : Is
IsSubmonoid.inter hS₁.to_isSubmonoid hS₂.to_isSubmonoid with }
#align is_subring.inter IsSubring.inter
-/- warning: is_subring.Inter -> IsSubring.interᵢ is a dubious translation:
+/- warning: is_subring.Inter -> IsSubring.iInter is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {ι : Sort.{u2}} {S : ι -> (Set.{u1} R)}, (forall (y : ι), IsSubring.{u1} R _inst_1 (S y)) -> (IsSubring.{u1} R _inst_1 (Set.interᵢ.{u1, u2} R ι S))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {ι : Sort.{u2}} {S : ι -> (Set.{u1} R)}, (forall (y : ι), IsSubring.{u1} R _inst_1 (S y)) -> (IsSubring.{u1} R _inst_1 (Set.iInter.{u1, u2} R ι S))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {ι : Sort.{u1}} {S : ι -> (Set.{u2} R)}, (forall (y : ι), IsSubring.{u2} R _inst_1 (S y)) -> (IsSubring.{u2} R _inst_1 (Set.interᵢ.{u2, u1} R ι S))
-Case conversion may be inaccurate. Consider using '#align is_subring.Inter IsSubring.interᵢₓ'. -/
-theorem IsSubring.interᵢ {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
- IsSubring (Set.interᵢ S) :=
- { IsAddSubgroup.interᵢ fun i => (h i).to_isAddSubgroup,
- IsSubmonoid.interᵢ fun i => (h i).to_isSubmonoid with }
-#align is_subring.Inter IsSubring.interᵢ
-
-/- warning: is_subring_Union_of_directed -> isSubring_unionᵢ_of_directed is a dubious translation:
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {ι : Sort.{u1}} {S : ι -> (Set.{u2} R)}, (forall (y : ι), IsSubring.{u2} R _inst_1 (S y)) -> (IsSubring.{u2} R _inst_1 (Set.iInter.{u2, u1} R ι S))
+Case conversion may be inaccurate. Consider using '#align is_subring.Inter IsSubring.iInterₓ'. -/
+theorem IsSubring.iInter {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
+ IsSubring (Set.iInter S) :=
+ { IsAddSubgroup.iInter fun i => (h i).to_isAddSubgroup,
+ IsSubmonoid.iInter fun i => (h i).to_isSubmonoid with }
+#align is_subring.Inter IsSubring.iInter
+
+/- warning: is_subring_Union_of_directed -> isSubring_iUnion_of_directed is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {ι : Type.{u2}} [hι : Nonempty.{succ u2} ι] {s : ι -> (Set.{u1} R)}, (forall (i : ι), IsSubring.{u1} R _inst_1 (s i)) -> (forall (i : ι) (j : ι), Exists.{succ u2} ι (fun (k : ι) => And (HasSubset.Subset.{u1} (Set.{u1} R) (Set.hasSubset.{u1} R) (s i) (s k)) (HasSubset.Subset.{u1} (Set.{u1} R) (Set.hasSubset.{u1} R) (s j) (s k)))) -> (IsSubring.{u1} R _inst_1 (Set.unionᵢ.{u1, succ u2} R ι (fun (i : ι) => s i)))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {ι : Type.{u2}} [hι : Nonempty.{succ u2} ι] {s : ι -> (Set.{u1} R)}, (forall (i : ι), IsSubring.{u1} R _inst_1 (s i)) -> (forall (i : ι) (j : ι), Exists.{succ u2} ι (fun (k : ι) => And (HasSubset.Subset.{u1} (Set.{u1} R) (Set.hasSubset.{u1} R) (s i) (s k)) (HasSubset.Subset.{u1} (Set.{u1} R) (Set.hasSubset.{u1} R) (s j) (s k)))) -> (IsSubring.{u1} R _inst_1 (Set.iUnion.{u1, succ u2} R ι (fun (i : ι) => s i)))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {ι : Type.{u1}} [hι : Nonempty.{succ u1} ι] {s : ι -> (Set.{u2} R)}, (forall (i : ι), IsSubring.{u2} R _inst_1 (s i)) -> (forall (i : ι) (j : ι), Exists.{succ u1} ι (fun (k : ι) => And (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s i) (s k)) (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s j) (s k)))) -> (IsSubring.{u2} R _inst_1 (Set.unionᵢ.{u2, succ u1} R ι (fun (i : ι) => s i)))
-Case conversion may be inaccurate. Consider using '#align is_subring_Union_of_directed isSubring_unionᵢ_of_directedₓ'. -/
-theorem isSubring_unionᵢ_of_directed {ι : Type _} [hι : Nonempty ι] {s : ι → Set R}
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {ι : Type.{u1}} [hι : Nonempty.{succ u1} ι] {s : ι -> (Set.{u2} R)}, (forall (i : ι), IsSubring.{u2} R _inst_1 (s i)) -> (forall (i : ι) (j : ι), Exists.{succ u1} ι (fun (k : ι) => And (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s i) (s k)) (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) (s j) (s k)))) -> (IsSubring.{u2} R _inst_1 (Set.iUnion.{u2, succ u1} R ι (fun (i : ι) => s i)))
+Case conversion may be inaccurate. Consider using '#align is_subring_Union_of_directed isSubring_iUnion_of_directedₓ'. -/
+theorem isSubring_iUnion_of_directed {ι : Type _} [hι : Nonempty ι] {s : ι → Set R}
(h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubring (⋃ i, s i) :=
- { to_isAddSubgroup := isAddSubgroup_unionᵢ_of_directed (fun i => (h i).to_isAddSubgroup) Directed
- to_isSubmonoid := isSubmonoid_unionᵢ_of_directed (fun i => (h i).to_isSubmonoid) Directed }
-#align is_subring_Union_of_directed isSubring_unionᵢ_of_directed
+ { to_isAddSubgroup := isAddSubgroup_iUnion_of_directed (fun i => (h i).to_isAddSubgroup) Directed
+ to_isSubmonoid := isSubmonoid_iUnion_of_directed (fun i => (h i).to_isSubmonoid) Directed }
+#align is_subring_Union_of_directed isSubring_iUnion_of_directed
namespace Ring
mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1
@@ -67,7 +67,7 @@ namespace RingHom
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_preimage RingHom.isSubring_preimageₓ'. -/
theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
@@ -79,7 +79,7 @@ theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_image RingHom.isSubring_imageₓ'. -/
theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
@@ -91,7 +91,7 @@ theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+*
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_2)) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S _inst_3)))))) f))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_set_range RingHom.isSubring_set_rangeₓ'. -/
theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
@@ -157,7 +157,7 @@ attribute [local reducible] closure
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.Mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.hasMem.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.Mem.{u1, u1} R (List.{u1} R) (List.hasMem.{u1} R) x l) -> (Or (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) L)) a)))
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.instMembershipList.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.mem.{u1, u1} R (List.{u1} R) (List.instMembershipList.{u1} R) x l) -> (Or (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) L)) a)))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.instMembershipList.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.mem.{u1, u1} R (List.{u1} R) (List.instMembershipList.{u1} R) x l) -> (Or (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))) L)) a)))
Case conversion may be inaccurate. Consider using '#align ring.exists_list_of_mem_closure Ring.exists_list_of_mem_closureₓ'. -/
theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
∃ L : List (List R), (∀ l ∈ L, ∀ x ∈ l, x ∈ s ∨ x = (-1 : R)) ∧ (L.map List.prod).Sum = a :=
@@ -188,7 +188,7 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) -> (C (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))))) -> (forall (z : R), (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y))) -> (C x)
but is expected to have type
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) -> (C (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) -> (forall (z : R), (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y))) -> (C x)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) -> (C (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) -> (forall (z : R), (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y))) -> (C x)
Case conversion may be inaccurate. Consider using '#align ring.in_closure.rec_on Ring.InClosure.recOnₓ'. -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -296,7 +296,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {S : Type.{u2}} [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (s : Set.{u1} R), Eq.{succ u2} (Set.{u2} S) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) (Ring.closure.{u1} R _inst_1 s)) (Ring.closure.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) s))
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring.image_closure Ring.image_closureₓ'. -/
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1
@@ -171,7 +171,7 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
match b, ih with
| _, ⟨L1, h1, rfl⟩ =>
⟨L1.map (List.cons (-1)), fun L2 h2 =>
- match L2, List.mem_map'.1 h2 with
+ match L2, List.mem_map.1 h2 with
| _, ⟨L3, h3, rfl⟩ => List.forall_mem_cons.2 ⟨Or.inr rfl, h1 L3 h3⟩,
by
simp only [List.map_map, (· ∘ ·), List.prod_cons, neg_one_mul] <;>
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce86f4e05e9a9b8da5e316b22c76ce76440c56a1
@@ -155,7 +155,7 @@ attribute [local reducible] closure
/- warning: ring.exists_list_of_mem_closure -> Ring.exists_list_of_mem_closure is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.Mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.hasMem.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.Mem.{u1, u1} R (List.{u1} R) (List.hasMem.{u1} R) x l) -> (Or (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) L)) a)))
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.Mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.hasMem.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.Mem.{u1, u1} R (List.{u1} R) (List.hasMem.{u1} R) x l) -> (Or (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1)) (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) L)) a)))
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {a : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) a (Ring.closure.{u1} R _inst_1 s)) -> (Exists.{succ u1} (List.{u1} (List.{u1} R)) (fun (L : List.{u1} (List.{u1} R)) => And (forall (l : List.{u1} R), (Membership.mem.{u1, u1} (List.{u1} R) (List.{u1} (List.{u1} R)) (List.instMembershipList.{u1} (List.{u1} R)) l L) -> (forall (x : R), (Membership.mem.{u1, u1} R (List.{u1} R) (List.instMembershipList.{u1} R) x l) -> (Or (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x s) (Eq.{succ u1} R x (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) (Eq.{succ u1} R (List.sum.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (List.map.{u1, u1} (List.{u1} R) R (List.prod.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) L)) a)))
Case conversion may be inaccurate. Consider using '#align ring.exists_list_of_mem_closure Ring.exists_list_of_mem_closureₓ'. -/
@@ -186,7 +186,7 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
/- warning: ring.in_closure.rec_on -> Ring.InClosure.recOn is a dubious translation:
lean 3 declaration is
- forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))))) -> (C (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (NonAssocRing.toAddGroupWithOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (forall (z : R), (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y))) -> (C x)
+ forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))))) -> (C (Neg.neg.{u1} R (SubNegMonoid.toHasNeg.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1))))))))) -> (forall (z : R), (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R _inst_1))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R _inst_1))) x y))) -> (C x)
but is expected to have type
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {s : Set.{u1} R} {C : R -> Prop} {x : R}, (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) x (Ring.closure.{u1} R _inst_1 s)) -> (C (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))) -> (C (Neg.neg.{u1} R (Ring.toNeg.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) -> (forall (z : R), (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) z s) -> (forall (n : R), (C n) -> (C (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) z n)))) -> (forall {x : R} {y : R}, (C x) -> (C y) -> (C (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))))) x y))) -> (C x)
Case conversion may be inaccurate. Consider using '#align ring.in_closure.rec_on Ring.InClosure.recOnₓ'. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212
@@ -67,7 +67,7 @@ namespace RingHom
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_preimage RingHom.isSubring_preimageₓ'. -/
theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
@@ -79,7 +79,7 @@ theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_image RingHom.isSubring_imageₓ'. -/
theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
@@ -91,7 +91,7 @@ theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+*
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_set_range RingHom.isSubring_set_rangeₓ'. -/
theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
@@ -296,7 +296,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {S : Type.{u2}} [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (s : Set.{u1} R), Eq.{succ u2} (Set.{u2} S) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) (Ring.closure.{u1} R _inst_1 s)) (Ring.closure.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) s))
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring.image_closure Ring.image_closureₓ'. -/
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212
@@ -250,12 +250,12 @@ theorem closure.isSubring : IsSubring (closure s) :=
(fun c hc =>
AddGroup.InClosure.rec_on ha
(fun d hd => AddGroup.subset_closure ((Monoid.closure.isSubmonoid _).mul_mem hd hc))
- ((zero_mul c).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
+ ((MulZeroClass.zero_mul c).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
(fun d hd hdc =>
neg_mul_eq_neg_mul d c ▸ (AddGroup.closure.isAddSubgroup _).neg_mem hdc)
fun d e hd he hdc hec =>
(add_mul d e c).symm ▸ (AddGroup.closure.isAddSubgroup _).add_mem hdc hec)
- ((mul_zero a).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
+ ((MulZeroClass.mul_zero a).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
(fun c hc hac => neg_mul_eq_mul_neg a c ▸ (AddGroup.closure.isAddSubgroup _).neg_mem hac)
fun c d hc hd hac had =>
(mul_add a c d).symm ▸ (AddGroup.closure.isAddSubgroup _).add_mem hac had }
mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5
@@ -67,7 +67,7 @@ namespace RingHom
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u2} S}, (IsSubring.{u2} S _inst_3 s) -> (IsSubring.{u1} R _inst_2 (Set.preimage.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_preimage RingHom.isSubring_preimageₓ'. -/
theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
(hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
@@ -79,7 +79,7 @@ theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) {s : Set.{u1} R}, (IsSubring.{u1} R _inst_2 s) -> (IsSubring.{u2} S _inst_3 (Set.image.{u1, u2} R S (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_image RingHom.isSubring_imageₓ'. -/
theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
(hs : IsSubring s) : IsSubring (f '' s) :=
@@ -91,7 +91,7 @@ theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+*
lean 3 declaration is
forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f))
but is expected to have type
- forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f))
+ forall {R : Type.{u1}} {S : Type.{u2}} [_inst_2 : Ring.{u1} R] [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))), IsSubring.{u2} S _inst_3 (Set.range.{u2, succ u1} S R (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)) (RingHom.instRingHomClassRingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_2)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3)))))) f))
Case conversion may be inaccurate. Consider using '#align ring_hom.is_subring_set_range RingHom.isSubring_set_rangeₓ'. -/
theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
IsSubring (Set.range f) :=
@@ -296,7 +296,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
lean 3 declaration is
forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R] {S : Type.{u2}} [_inst_3 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (s : Set.{u1} R), Eq.{succ u2} (Set.{u2} S) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) (Ring.closure.{u1} R _inst_1 s)) (Ring.closure.{u2} S _inst_3 (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_3))) f) s))
but is expected to have type
- forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2398 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) s))
+ forall {R : Type.{u2}} [_inst_1 : Ring.{u2} R] {S : Type.{u1}} [_inst_3 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (s : Set.{u2} R), Eq.{succ u1} (Set.{u1} S) (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) (Ring.closure.{u2} R _inst_1 s)) (Ring.closure.{u1} S _inst_3 (Set.image.{u2, u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2372 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_3)))))) f) s))
Case conversion may be inaccurate. Consider using '#align ring.image_closure Ring.image_closureₓ'. -/
theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/3ade05ac9447ae31a22d2ea5423435e054131240
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
! This file was ported from Lean 3 source module deprecated.subring
-! leanprover-community/mathlib commit 3d7987cda72abc473c7cdbbb075170e9ac620042
+! leanprover-community/mathlib commit 23aa88e32dcc9d2a24cca7bc23268567ed4cd7d6
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.Subring.Basic
/-!
# Unbundled subrings (deprecated)
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
This file is deprecated, and is no longer imported by anything in mathlib other than other
deprecated files, and test files. You should not need to import it.
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -79,13 +79,13 @@ theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : Is
IsSubmonoid.inter hS₁.toIsSubmonoid hS₂.toIsSubmonoid with }
#align is_subring.inter IsSubring.inter
-theorem IsSubring.iInter {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
+theorem IsSubring.iInter {ι : Sort*} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
IsSubring (Set.iInter S) :=
{ IsAddSubgroup.iInter fun i ↦ (h i).toIsAddSubgroup,
IsSubmonoid.iInter fun i ↦ (h i).toIsSubmonoid with }
#align is_subring.Inter IsSubring.iInter
-theorem isSubring_iUnion_of_directed {ι : Type _} [Nonempty ι] {s : ι → Set R}
+theorem isSubring_iUnion_of_directed {ι : Type*} [Nonempty ι] {s : ι → Set R}
(h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubring (⋃ i, s i) :=
{ toIsAddSubgroup := isAddSubgroup_iUnion_of_directed (fun i ↦ (h i).toIsAddSubgroup) directed
@@ -202,7 +202,7 @@ theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
closure_subset closure.isSubring <| Set.Subset.trans H subset_closure
#align ring.closure_mono Ring.closure_mono
-theorem image_closure {S : Type _} [Ring S] (f : R →+* S) (s : Set R) :
+theorem image_closure {S : Type*} [Ring S] (f : R →+* S) (s : Set R) :
f '' closure s = closure (f '' s) := by
refine' le_antisymm _ (closure_subset (RingHom.isSubring_image _ closure.isSubring) <|
Set.image_subset _ subset_closure)
@@ -2,16 +2,13 @@
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module deprecated.subring
-! leanprover-community/mathlib commit 2738d2ca56cbc63be80c3bd48e9ed90ad94e947d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Deprecated.Subgroup
import Mathlib.Deprecated.Group
import Mathlib.RingTheory.Subring.Basic
+#align_import deprecated.subring from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
+
/-!
# Unbundled subrings (deprecated)
sSup
/iSup
(#3938)
As discussed on Zulip
supₛ
→ sSup
infₛ
→ sInf
supᵢ
→ iSup
infᵢ
→ iInf
bsupₛ
→ bsSup
binfₛ
→ bsInf
bsupᵢ
→ biSup
binfᵢ
→ biInf
csupₛ
→ csSup
cinfₛ
→ csInf
csupᵢ
→ ciSup
cinfᵢ
→ ciInf
unionₛ
→ sUnion
interₛ
→ sInter
unionᵢ
→ iUnion
interᵢ
→ iInter
bunionₛ
→ bsUnion
binterₛ
→ bsInter
bunionᵢ
→ biUnion
binterᵢ
→ biInter
Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>
@@ -82,18 +82,18 @@ theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : Is
IsSubmonoid.inter hS₁.toIsSubmonoid hS₂.toIsSubmonoid with }
#align is_subring.inter IsSubring.inter
-theorem IsSubring.interᵢ {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
- IsSubring (Set.interᵢ S) :=
- { IsAddSubgroup.interᵢ fun i ↦ (h i).toIsAddSubgroup,
- IsSubmonoid.interᵢ fun i ↦ (h i).toIsSubmonoid with }
-#align is_subring.Inter IsSubring.interᵢ
+theorem IsSubring.iInter {ι : Sort _} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
+ IsSubring (Set.iInter S) :=
+ { IsAddSubgroup.iInter fun i ↦ (h i).toIsAddSubgroup,
+ IsSubmonoid.iInter fun i ↦ (h i).toIsSubmonoid with }
+#align is_subring.Inter IsSubring.iInter
-theorem isSubring_unionᵢ_of_directed {ι : Type _} [Nonempty ι] {s : ι → Set R}
+theorem isSubring_iUnion_of_directed {ι : Type _} [Nonempty ι] {s : ι → Set R}
(h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
IsSubring (⋃ i, s i) :=
- { toIsAddSubgroup := isAddSubgroup_unionᵢ_of_directed (fun i ↦ (h i).toIsAddSubgroup) directed
- toIsSubmonoid := isSubmonoid_unionᵢ_of_directed (fun i ↦ (h i).toIsSubmonoid) directed }
-#align is_subring_Union_of_directed isSubring_unionᵢ_of_directed
+ { toIsAddSubgroup := isAddSubgroup_iUnion_of_directed (fun i ↦ (h i).toIsAddSubgroup) directed
+ toIsSubmonoid := isSubmonoid_iUnion_of_directed (fun i ↦ (h i).toIsSubmonoid) directed }
+#align is_subring_Union_of_directed isSubring_iUnion_of_directed
namespace Ring
Notably incorporates https://github.com/leanprover/std4/pull/98 and https://github.com/leanprover/std4/pull/109.
https://github.com/leanprover/std4/pull/98 moves a number of lemmas from Mathlib to Std, so the bump requires deleting them in Mathlib. I did check on each lemma whether its attributes were kept in the move (and gave attribute markings in Mathlib if they were not present in Std), but a reviewer may wish to re-check.
List.mem_map
changed statement from b ∈ l.map f ↔ ∃ a, a ∈ l ∧ b = f a
to b ∈ l.map f ↔ ∃ a, a ∈ l ∧ f a = b
. Similarly for List.exists_of_mem_map
. This was a deliberate change, so I have simply adjusted proofs (many become simpler, which supports the change). I also deleted List.mem_map'
, List.exists_of_mem_map'
, which were temporary versions in Mathlib while waiting for this change (replacing their uses with the unprimed versions).
Also, the lemma sublist_nil_iff_eq_nil
seems to have been renamed to sublist_nil
during the move, so I added an alias for the old name.
(another issue fixed during review by @digama0) List.Sublist.filter
had an argument change from explicit to implicit. This appears to have been an oversight (cc @JamesGallicchio). I have temporarily introduced List.Sublist.filter'
with the argument explicit, and replaced Mathlib uses of Sublist.filter
with Sublist.filter'
. Later we can fix the argument in Std, and then delete List.Sublist.filter'
.
@@ -116,7 +116,7 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
fun {b} _ ih ↦ match b, ih with
| _, ⟨L1, h1, rfl⟩ =>
⟨L1.map (List.cons (-1)),
- fun L2 h2 ↦ match L2, List.mem_map'.1 h2 with
+ fun L2 h2 ↦ match L2, List.mem_map.1 h2 with
| _, ⟨L3, h3, rfl⟩ => List.forall_mem_cons.2 ⟨Or.inr rfl, h1 L3 h3⟩, by
simp only [List.map_map, (· ∘ ·), List.prod_cons, neg_one_mul]
refine' List.recOn L1 neg_zero.symm fun hd tl ih ↦ _
@@ -28,7 +28,7 @@ of the ring `R`. The bundled variant `Subring R` should be used in preference to
## Tags
-is_subring
+IsSubring
-/
The unported dependencies are