Module: Mathlib.LinearAlgebra.Eigenspace.Triangularizable

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
p : Submodule K V
f : End K V
inst✝ : FiniteDimensional K V
h : ∀ x ∈ p, f x ∈ p
k : ℕ∞
m : V
hm : m ∈ p ⊓ ⨆ μ, (f.genEigenspace μ) k
⊢ m ∈ ⨆ μ, p ⊓ (f.genEigenspace μ) k

Tactic:
classical
  obtain ⟨hm₀ : m ∈ p, hm₁ : m ∈ ⨆ μ, f.genEigenspace 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.notMem_support_iff.mp hμ, p.zero_mem]
  have hm₂_aux := hm₂
  simp_rw [Module.End.mem_genEigenspace] at hm₂_aux
  choose l hlk hl using hm₂_aux
  let l₀ : ℕ := m.support.sup l
  have h_comm : ∀ (μ₁ μ₂ : K),
    Commute ((f - algebraMap K (End K V) μ₁) ^ l₀)
            ((f - algebraMap K (End K V) μ₂) ^ l₀) := 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_finsuppSum, 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']
    have hl₀ : ((f - algebraMap K (End K V) μ') ^ l₀) (m μ') = 0 := by
      rw [← LinearMap.mem_ker, Algebra.algebraMap_eq_smul_one, ← End.mem_genEigenspace_nat]
      simp_rw [← End.mem_genEigenspace_nat] at hl
      suffices (l μ' : ℕ∞) ≤ l₀ from (f.genEigenspace μ').mono this (hl μ')
      simpa only [Nat.cast_le] using Finset.le_sup hμ'
    have : _ = g := (m.support.erase μ).noncommProd_erase_mul (Finset.mem_erase.mpr ⟨hμμ', hμ'⟩)
      (fun μ ↦ (f - algebraMap K (End K V) μ) ^ l₀) (fun μ₁ _ μ₂ _ _ ↦ h_comm μ₁ μ₂)
    rw [← this, Module.End.mul_apply, hl₀, _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 [Module.End.coe_pow, this.iterate l₀]
    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 ↑(f.genEigenspace μ k) ↑(f.genEigenspace μ k) :=
    f.mapsTo_genEigenspace_of_comm hfg μ k
  have hg₃ : InjOn g ↑(f.genEigenspace μ k) := by
    apply LinearMap.injOn_of_disjoint_ker subset_rfl
    have := f.independent_genEigenspace k
    have aux (μ') (_hμ' : μ' ∈ m.support.erase μ) :
        (f.genEigenspace μ') ↑l₀ ≤ (f.genEigenspace μ') k := by
      apply (f.genEigenspace μ').mono
      obtain _ | k := k
      · exact le_top
      · exact Nat.cast_le.2 <| Finset.sup_le fun i _ ↦ Nat.cast_le.1 <| hlk i
    rw [LinearMap.ker_noncommProd_eq_of_supIndep_ker, ← Finset.sup_eq_iSup]
    · have := Finset.supIndep_iff_disjoint_erase.mp (this.supIndep' m.support) μ hμ
      apply this.mono_right
      apply Finset.sup_mono_fun
      intro μ' hμ'
      rw [Algebra.algebraMap_eq_smul_one, ← End.genEigenspace_nat]
      apply aux μ' hμ'
    · have := this.supIndep' (m.support.erase μ)
      apply this.antitone_fun
      intro μ' hμ'
      rw [Algebra.algebraMap_eq_smul_one, ← End.genEigenspace_nat]
      apply aux μ' hμ'
  have hg₄ : SurjOn g
      ↑(p ⊓ f.genEigenspace μ k) ↑(p ⊓ f.genEigenspace μ k) := by
    have : MapsTo g
        ↑(p ⊓ f.genEigenspace μ k) ↑(p ⊓ f.genEigenspace 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 ∈ f.genEigenspace μ k⟩, hy₂ : g y = g (m μ)⟩ :=
    hg₄ ⟨(hg₀ ▸ hg₁ hm₀), hg₂ hm₂⟩
  rwa [← hg₃ hy₁ hm₂ hy₂]

Goal state after: