linear_algebra.eigenspace.is_alg_closed ⟷ Mathlib.LinearAlgebra.Eigenspace.Triangularizable

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -64,7 +64,7 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
   cases n
   -- If the vector space is 0-dimensional, the result is trivial.
   · rw [← top_le_iff]
-    simp only [finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]
+    simp only [Submodule.finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]
   -- Otherwise the vector space is nontrivial.
   · haveI : Nontrivial V := finrank_pos_iff.1 (by rw [h_dim]; apply Nat.zero_lt_succ)
     -- Hence, `f` has an eigenvalue `μ₀`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,8 +3,8 @@ Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 -/
-import Mathbin.LinearAlgebra.Eigenspace.Basic
-import Mathbin.FieldTheory.IsAlgClosed.Spectrum
+import LinearAlgebra.Eigenspace.Basic
+import FieldTheory.IsAlgClosed.Spectrum
 
 #align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"660b3a2db3522fa0db036e569dc995a615c4c848"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
-
-! This file was ported from Lean 3 source module linear_algebra.eigenspace.is_alg_closed
-! leanprover-community/mathlib commit 660b3a2db3522fa0db036e569dc995a615c4c848
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.LinearAlgebra.Eigenspace.Basic
 import Mathbin.FieldTheory.IsAlgClosed.Spectrum
 
+#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"660b3a2db3522fa0db036e569dc995a615c4c848"
+
 /-!
 # Eigenvectors and eigenvalues over algebraically closed fields.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -57,6 +57,7 @@ noncomputable instance [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f
     Inhabited f.Eigenvalues :=
   ⟨⟨f.exists_eigenvalue.some, f.exists_eigenvalue.choose_spec⟩⟩
 
+#print Module.End.iSup_generalizedEigenspace_eq_top /-
 /-- The generalized eigenvectors span the entire vector space (Lemma 8.21 of [axler2015]). -/
 theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
     (⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k) = ⊤ :=
@@ -122,6 +123,7 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
     · rw [← top_le_iff, ← Submodule.eq_top_of_disjoint ER ES h_dim_add h_disjoint]
       apply sup_le hER hES
 #align module.End.supr_generalized_eigenspace_eq_top Module.End.iSup_generalizedEigenspace_eq_top
+-/
 
 end End
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
 
 ! This file was ported from Lean 3 source module linear_algebra.eigenspace.is_alg_closed
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
+! leanprover-community/mathlib commit 660b3a2db3522fa0db036e569dc995a615c4c848
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.FieldTheory.IsAlgClosed.Spectrum
 /-!
 # Eigenvectors and eigenvalues over algebraically closed fields.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 * Every linear operator on a vector space over an algebraically closed field has an eigenvalue.
 * The generalized eigenvectors span the entire vector space.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -38,6 +38,7 @@ open FiniteDimensional
 
 variable {K : Type v} {V : Type w} [Field K] [AddCommGroup V] [Module K V]
 
+#print Module.End.exists_eigenvalue /-
 -- This is Lemma 5.21 of [axler2015], although we are no longer following that proof.
 /-- Every linear operator on a vector space over an algebraically closed field has
     an eigenvalue. -/
@@ -47,6 +48,7 @@ theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V]
   simp_rw [has_eigenvalue_iff_mem_spectrum]
   exact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f
 #align module.End.exists_eigenvalue Module.End.exists_eigenvalue
+-/
 
 noncomputable instance [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
     Inhabited f.Eigenvalues :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -140,7 +140,8 @@ theorem inf_iSup_generalizedEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p,
   suffices ∀ μ, (m μ : V) ∈ p by
     exact (mem_iSup_iff_exists_finsupp _ _).mpr ⟨m, fun μ ↦ mem_inf.mp ⟨this μ, hm₂ μ⟩, rfl⟩
   intro μ
-  by_cases hμ : μ ∈ m.support; swap; simp only [Finsupp.not_mem_support_iff.mp hμ, p.zero_mem]
+  by_cases hμ : μ ∈ m.support; swap
+  · simp only [Finsupp.not_mem_support_iff.mp hμ, p.zero_mem]
   have h_comm : ∀ (μ₁ μ₂ : K),
     Commute ((f - algebraMap K (End K V) μ₁) ^ finrank K V)
             ((f - algebraMap K (End K V) μ₂) ^ finrank K V) := fun μ₁ μ₂ ↦

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -84,7 +84,7 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
     let f' : End K ER := f.restrict h_f_ER
     -- The dimension of `ES` is positive
     have h_dim_ES_pos : 0 < finrank K ES := by
-      dsimp only
+      dsimp only [ES]
       rw [h_dim]
       apply pos_finrank_generalizedEigenspace_of_hasEigenvalue hμ₀ (Nat.zero_lt_succ n)
     -- and the dimensions of `ES` and `ER` add up to `finrank K V`.

A portion of results in Mathlib/LinearAlgebra/FiniteDimensional.lean were generalized to rings and moved to Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean. Most API lemmas for FiniteDimensional are kept but replaced with one lemma proofs. Definitions and niche lemmas are replaced by the generalized version completely.

Co-authored-by: erd1 <the.erd.one@gmail.com> Co-authored-by: Andrew Yang <the.erd.one@gmail.com>

@@ -68,7 +68,7 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
   cases' n with n
   -- If the vector space is 0-dimensional, the result is trivial.
   · rw [← top_le_iff]
-    simp only [finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]
+    simp only [Submodule.finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]
   -- Otherwise the vector space is nontrivial.
   · haveI : Nontrivial V := finrank_pos_iff.1 (by rw [h_dim]; apply Nat.zero_lt_succ)
     -- Hence, `f` has an eigenvalue `μ₀`.

@@ -154,7 +154,8 @@ theorem inf_iSup_generalizedEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p,
       rw [map_finsupp_sum, Finsupp.sum_congr (g2 := fun μ' _ ↦ if μ' = μ then g (m μ) else 0) this,
         Finsupp.sum_ite_eq', if_pos hμ]
     rintro μ' hμ'
-    split_ifs with hμμ'; rw [hμμ']
+    split_ifs with hμμ'
+    · rw [hμμ']
     replace hm₂ : ((f - algebraMap K (End K V) μ') ^ finrank K V) (m μ') = 0 := by
       obtain ⟨k, hk⟩ := (mem_iSup_of_chain _ _).mp (hm₂ μ')
       exact Module.End.generalizedEigenspace_le_generalizedEigenspace_finrank _ _ k hk

@@ -13,33 +13,41 @@ import Mathlib.FieldTheory.IsAlgClosed.Spectrum
 
 This file contains basic results relevant to the triangularizability of linear endomorphisms.
 
-* Every linear operator on a vector space over an algebraically closed field has an eigenvalue.
-* The generalized eigenvectors span the entire vector space.
+## Main definitions / results
+
+ * `Module.End.exists_eigenvalue`: in finite dimensions, over an algebraically closed field, every
+   linear endomorphism has an eigenvalue.
+ * `Module.End.iSup_generalizedEigenspace_eq_top`: in finite dimensions, over an algebraically
+   closed field, the generalized eigenspaces of any linear endomorphism span the whole space.
+ * `Module.End.iSup_generalizedEigenspace_restrict_eq_top`: in finite dimensions, if the
+   generalized eigenspaces of a linear endomorphism span the whole space then the same is true of
+   its restriction to any invariant submodule.
 
 ## References
 
 * [Sheldon Axler, *Linear Algebra Done Right*][axler2015]
 * https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors
 
+## TODO
+
+Define triangularizable endomorphisms (e.g., as existence of a maximal chain of invariant subspaces)
+and prove that in finite dimensions over a field, this is equivalent to the property that the
+generalized eigenspaces span the whole space.
+
 ## Tags
 
 eigenspace, eigenvector, eigenvalue, eigen
 -/
 
+open Set Function Module FiniteDimensional
 
-universe u v w
-
-namespace Module
+variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
 
-namespace End
-
-open FiniteDimensional
-
-variable {K : Type v} {V : Type w} [Field K] [AddCommGroup V] [Module K V]
+namespace Module.End
 
 -- This is Lemma 5.21 of [axler2015], although we are no longer following that proof.
-/-- Every linear operator on a vector space over an algebraically closed field has
-    an eigenvalue. -/
+/-- In finite dimensions, over an algebraically closed field, every linear endomorphism has an
+eigenvalue. -/
 theorem exists_eigenvalue [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f : End K V) :
     ∃ c : K, f.HasEigenvalue c := by
   simp_rw [hasEigenvalue_iff_mem_spectrum]
@@ -50,7 +58,9 @@ noncomputable instance [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f
     Inhabited f.Eigenvalues :=
   ⟨⟨f.exists_eigenvalue.choose, f.exists_eigenvalue.choose_spec⟩⟩
 
-/-- The generalized eigenvectors span the entire vector space (Lemma 8.21 of [axler2015]). -/
+-- Lemma 8.21 of [axler2015]
+/-- In finite dimensions, over an algebraically closed field, the generalized eigenspaces of any
+linear endomorphism span the whole space. -/
 theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
     ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k = ⊤ := by
   -- We prove the claim by strong induction on the dimension of the vector space.
@@ -113,6 +123,87 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
       apply sup_le hER hES
 #align module.End.supr_generalized_eigenspace_eq_top Module.End.iSup_generalizedEigenspace_eq_top
 
-end End
-
-end Module
+end Module.End
+
+namespace Submodule
+
+variable {p : Submodule K V} {f : Module.End K V}
+
+theorem inf_iSup_generalizedEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) :
+    p ⊓ ⨆ μ, ⨆ k, f.generalizedEigenspace μ k = ⨆ μ, ⨆ k, p ⊓ f.generalizedEigenspace μ k := by
+  simp_rw [← (f.generalizedEigenspace _).mono.directed_le.inf_iSup_eq]
+  refine le_antisymm (fun m hm ↦ ?_)
+    (le_inf_iff.mpr ⟨iSup_le fun μ ↦ inf_le_left, iSup_mono fun μ ↦ inf_le_right⟩)
+  classical
+  obtain ⟨hm₀ : m ∈ p, hm₁ : m ∈ ⨆ μ, ⨆ k, f.generalizedEigenspace μ k⟩ := hm
+  obtain ⟨m, hm₂, rfl⟩ := (mem_iSup_iff_exists_finsupp _ _).mp hm₁
+  suffices ∀ μ, (m μ : V) ∈ p by
+    exact (mem_iSup_iff_exists_finsupp _ _).mpr ⟨m, fun μ ↦ mem_inf.mp ⟨this μ, hm₂ μ⟩, rfl⟩
+  intro μ
+  by_cases hμ : μ ∈ m.support; swap; simp only [Finsupp.not_mem_support_iff.mp hμ, p.zero_mem]
+  have h_comm : ∀ (μ₁ μ₂ : K),
+    Commute ((f - algebraMap K (End K V) μ₁) ^ finrank K V)
+            ((f - algebraMap K (End K V) μ₂) ^ finrank K V) := fun μ₁ μ₂ ↦
+    ((Commute.sub_right rfl <| Algebra.commute_algebraMap_right _ _).sub_left
+      (Algebra.commute_algebraMap_left _ _)).pow_pow _ _
+  let g : End K V := (m.support.erase μ).noncommProd _ fun μ₁ _ μ₂ _ _ ↦ h_comm μ₁ μ₂
+  have hfg : Commute f g := Finset.noncommProd_commute _ _ _ _ fun μ' _ ↦
+    (Commute.sub_right rfl <| Algebra.commute_algebraMap_right _ _).pow_right _
+  have hg₀ : g (m.sum fun _μ mμ ↦ mμ) = g (m μ) := by
+    suffices ∀ μ' ∈ m.support, g (m μ') = if μ' = μ then g (m μ) else 0 by
+      rw [map_finsupp_sum, Finsupp.sum_congr (g2 := fun μ' _ ↦ if μ' = μ then g (m μ) else 0) this,
+        Finsupp.sum_ite_eq', if_pos hμ]
+    rintro μ' hμ'
+    split_ifs with hμμ'; rw [hμμ']
+    replace hm₂ : ((f - algebraMap K (End K V) μ') ^ finrank K V) (m μ') = 0 := by
+      obtain ⟨k, hk⟩ := (mem_iSup_of_chain _ _).mp (hm₂ μ')
+      exact Module.End.generalizedEigenspace_le_generalizedEigenspace_finrank _ _ k hk
+    have : _ = g := (m.support.erase μ).noncommProd_erase_mul (Finset.mem_erase.mpr ⟨hμμ', hμ'⟩)
+      (fun μ ↦ (f - algebraMap K (End K V) μ) ^ finrank K V) (fun μ₁ _ μ₂ _ _ ↦ h_comm μ₁ μ₂)
+    rw [← this, LinearMap.mul_apply, hm₂, _root_.map_zero]
+  have hg₁ : MapsTo g p p := Finset.noncommProd_induction _ _ _ (fun g' : End K V ↦ MapsTo g' p p)
+      (fun f₁ f₂ ↦ MapsTo.comp) (mapsTo_id _) fun μ' _ ↦ by
+    suffices MapsTo (f - algebraMap K (End K V) μ') p p by
+      simp only [LinearMap.coe_pow]; exact this.iterate (finrank K V)
+    intro x hx
+    rw [LinearMap.sub_apply, algebraMap_end_apply]
+    exact p.sub_mem (h _ hx) (smul_mem p μ' hx)
+  have hg₂ : MapsTo g ↑(⨆ k, f.generalizedEigenspace μ k) ↑(⨆ k, f.generalizedEigenspace μ k) :=
+    f.mapsTo_iSup_generalizedEigenspace_of_comm hfg μ
+  have hg₃ : InjOn g ↑(⨆ k, f.generalizedEigenspace μ k) := by
+    apply LinearMap.injOn_of_disjoint_ker (subset_refl _)
+    have this := f.independent_generalizedEigenspace
+    simp_rw [f.iSup_generalizedEigenspace_eq_generalizedEigenspace_finrank] at this ⊢
+    rw [LinearMap.ker_noncommProd_eq_of_supIndep_ker _ _ <| this.supIndep' (m.support.erase μ),
+      ← Finset.sup_eq_iSup]
+    exact Finset.supIndep_iff_disjoint_erase.mp (this.supIndep' m.support) μ hμ
+  have hg₄ : SurjOn g
+      ↑(p ⊓ ⨆ k, f.generalizedEigenspace μ k) ↑(p ⊓ ⨆ k, f.generalizedEigenspace μ k) := by
+    have : MapsTo g
+        ↑(p ⊓ ⨆ k, f.generalizedEigenspace μ k) ↑(p ⊓ ⨆ k, f.generalizedEigenspace μ k) :=
+      hg₁.inter_inter hg₂
+    rw [← LinearMap.injOn_iff_surjOn this]
+    exact hg₃.mono (inter_subset_right _ _)
+  specialize hm₂ μ
+  obtain ⟨y, ⟨hy₀ : y ∈ p, hy₁ : y ∈ ⨆ k, f.generalizedEigenspace μ k⟩, hy₂ : g y = g (m μ)⟩ :=
+    hg₄ ⟨(hg₀ ▸ hg₁ hm₀), hg₂ hm₂⟩
+  rwa [← hg₃ hy₁ hm₂ hy₂]
+
+theorem eq_iSup_inf_generalizedEigenspace [FiniteDimensional K V]
+    (h : ∀ x ∈ p, f x ∈ p) (h' : ⨆ μ, ⨆ k, f.generalizedEigenspace μ k = ⊤) :
+    p = ⨆ μ, ⨆ k, p ⊓ f.generalizedEigenspace μ k := by
+  rw [← inf_iSup_generalizedEigenspace h, h', inf_top_eq]
+
+end Submodule
+
+/-- In finite dimensions, if the generalized eigenspaces of a linear endomorphism span the whole
+space then the same is true of its restriction to any invariant submodule. -/
+theorem Module.End.iSup_generalizedEigenspace_restrict_eq_top
+    {p : Submodule K V} {f : Module.End K V} [FiniteDimensional K V]
+    (h : ∀ x ∈ p, f x ∈ p) (h' : ⨆ μ, ⨆ k, f.generalizedEigenspace μ k = ⊤) :
+    ⨆ μ, ⨆ k, Module.End.generalizedEigenspace (LinearMap.restrict f h) μ k = ⊤ := by
+  have := congr_arg (Submodule.comap p.subtype) (Submodule.eq_iSup_inf_generalizedEigenspace h h')
+  have h_inj : Function.Injective p.subtype := Subtype.coe_injective
+  simp_rw [Submodule.inf_generalizedEigenspace f p h, Submodule.comap_subtype_self,
+    ← Submodule.map_iSup, Submodule.comap_map_eq_of_injective h_inj] at this
+  exact this.symm

The motivation is that I wish to add further results about triangularizability of linear endomorphisms and I claim this file is a natural home.

This rename-only PR exists to prevent breaking 'nightly-testing'.

@@ -9,7 +9,9 @@ import Mathlib.FieldTheory.IsAlgClosed.Spectrum
 #align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
 
 /-!
-# Eigenvectors and eigenvalues over algebraically closed fields.
+# Triangularizable linear endomorphisms
+
+This file contains basic results relevant to the triangularizability of linear endomorphisms.
 
 * Every linear operator on a vector space over an algebraically closed field has an eigenvalue.
 * The generalized eigenvectors span the entire vector space.

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,15 +2,12 @@
 Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alexander Bentkamp
-
-! This file was ported from Lean 3 source module linear_algebra.eigenspace.is_alg_closed
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.LinearAlgebra.Eigenspace.Basic
 import Mathlib.FieldTheory.IsAlgClosed.Spectrum
 
+#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
+
 /-!
 # Eigenvectors and eigenvalues over algebraically closed fields.
 

@@ -53,7 +53,7 @@ noncomputable instance [IsAlgClosed K] [FiniteDimensional K V] [Nontrivial V] (f
 
 /-- The generalized eigenvectors span the entire vector space (Lemma 8.21 of [axler2015]). -/
 theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : End K V) :
-    (⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k) = ⊤ := by
+    ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k = ⊤ := by
   -- We prove the claim by strong induction on the dimension of the vector space.
   induction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V
   cases' n with n
@@ -84,11 +84,11 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
     -- Therefore the dimension `ER` mus be smaller than `finrank K V`.
     have h_dim_ER : finrank K ER < n.succ := by linarith
     -- This allows us to apply the induction hypothesis on `ER`:
-    have ih_ER : (⨆ (μ : K) (k : ℕ), f'.generalizedEigenspace μ k) = ⊤ :=
+    have ih_ER : ⨆ (μ : K) (k : ℕ), f'.generalizedEigenspace μ k = ⊤ :=
       ih (finrank K ER) h_dim_ER f' rfl
     -- The induction hypothesis gives us a statement about subspaces of `ER`. We can transfer this
     -- to a statement about subspaces of `V` via `submodule.subtype`:
-    have ih_ER' : (⨆ (μ : K) (k : ℕ), (f'.generalizedEigenspace μ k).map ER.subtype) = ER := by
+    have ih_ER' : ⨆ (μ : K) (k : ℕ), (f'.generalizedEigenspace μ k).map ER.subtype = ER := by
       simp only [(Submodule.map_iSup _ _).symm, ih_ER, Submodule.map_subtype_top ER]
     -- Moreover, every generalized eigenspace of `f'` is contained in the corresponding generalized
     -- eigenspace of `f`.
@@ -109,7 +109,7 @@ theorem iSup_generalizedEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V
     have h_disjoint : Disjoint ER ES := generalized_eigenvec_disjoint_range_ker f μ₀
     -- Since the dimensions of `ER` and `ES` add up to the dimension of `V`, it follows that the
     -- span of all generalized eigenvectors is all of `V`.
-    show (⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k) = ⊤
+    show ⨆ (μ : K) (k : ℕ), f.generalizedEigenspace μ k = ⊤
     · rw [← top_le_iff, ← Submodule.eq_top_of_disjoint ER ES h_dim_add h_disjoint]
       apply sup_le hER hES
 #align module.End.supr_generalized_eigenspace_eq_top Module.End.iSup_generalizedEigenspace_eq_top