Theorem: Module.End.exists_ker_pow_eq_ker_pow_succ
File: Mathlib/LinearAlgebra/FiniteDimensional.lean

tactic: by_contra h_contra
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
⊢ ∃ k ≤ finrank K V, LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ¬∃ k ≤ finrank K V, LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
⊢ False

tactic: simp_rw [not_exists, not_and] at h_contra
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ¬∃ k ≤ finrank K V, LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)
⊢ False
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
⊢ False

tactic: have h_any_n_lt : ∀ n, n ≤ (finrank K V).succ → n ≤ finrank K V := fun n hn =>
  (h_le_ker_pow n hn).trans (Submodule.finrank_le _)
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
h_le_ker_pow : ∀ n ≤ (finrank K V).succ, n ≤ finrank K ↥(LinearMap.ker (f ^ n))
⊢ False
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
h_le_ker_pow : ∀ n ≤ (finrank K V).succ, n ≤ finrank K ↥(LinearMap.ker (f ^ n))
h_any_n_lt : ∀ n ≤ (finrank K V).succ, n ≤ finrank K V
⊢ False

tactic: exact Nat.not_succ_le_self _ (h_any_n_lt (finrank K V).succ (finrank K V).succ.le_refl)
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
h_le_ker_pow : ∀ n ≤ (finrank K V).succ, n ≤ finrank K ↥(LinearMap.ker (f ^ n))
h_any_n_lt : ∀ n ≤ (finrank K V).succ, n ≤ finrank K V
⊢ False
state_after: no goals

tactic: intro n hn
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
⊢ ∀ n ≤ (finrank K V).succ, n ≤ finrank K ↥(LinearMap.ker (f ^ n))
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
hn : n ≤ (finrank K V).succ
⊢ n ≤ finrank K ↥(LinearMap.ker (f ^ n))

tactic: induction' n with n ih
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
hn : n ≤ (finrank K V).succ
⊢ n ≤ finrank K ↥(LinearMap.ker (f ^ n))
state_after: case zero
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
hn : 0 ≤ (finrank K V).succ
⊢ 0 ≤ finrank K ↥(LinearMap.ker (f ^ 0))

case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))

tactic: exact zero_le (finrank _ _)
state_before: case zero
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
hn : 0 ≤ (finrank K V).succ
⊢ 0 ≤ finrank K ↥(LinearMap.ker (f ^ 0))
state_after: no goals

tactic: have h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ) := by
  refine lt_of_le_of_ne ?_ (h_contra n (Nat.le_of_succ_le_succ hn))
  rw [pow_succ']
  apply LinearMap.ker_le_ker_comp
state_before: case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))
state_after: case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))

tactic: have h_finrank_lt_finrank :
    finrank K (LinearMap.ker (f ^ n)) < finrank K (LinearMap.ker (f ^ n.succ)) := by
  apply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker
state_before: case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))
state_after: case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
h_finrank_lt_finrank : finrank K ↥(LinearMap.ker (f ^ n)) < finrank K ↥(LinearMap.ker (f ^ n.succ))
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))

tactic: calc
  n.succ ≤ (finrank K ↑(LinearMap.ker (f ^ n))).succ :=
    Nat.succ_le_succ (ih (Nat.le_of_succ_le hn))
  _ ≤ finrank K ↑(LinearMap.ker (f ^ n.succ)) := Nat.succ_le_of_lt h_finrank_lt_finrank
state_before: case succ
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
h_finrank_lt_finrank : finrank K ↥(LinearMap.ker (f ^ n)) < finrank K ↥(LinearMap.ker (f ^ n.succ))
⊢ n + 1 ≤ finrank K ↥(LinearMap.ker (f ^ (n + 1)))
state_after: no goals

tactic: refine lt_of_le_of_ne ?_ (h_contra n (Nat.le_of_succ_le_succ hn))
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ LinearMap.ker (f ^ n) ≤ LinearMap.ker (f ^ n.succ)

tactic: rw [pow_succ']
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ LinearMap.ker (f ^ n) ≤ LinearMap.ker (f ^ n.succ)
state_after: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ LinearMap.ker (f ^ n) ≤ LinearMap.ker (f * f ^ n)

tactic: apply LinearMap.ker_le_ker_comp
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
⊢ LinearMap.ker (f ^ n) ≤ LinearMap.ker (f * f ^ n)
state_after: no goals

tactic: apply Submodule.finrank_lt_finrank_of_lt h_ker_lt_ker
state_before: K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : End K V
h_contra : ∀ x ≤ finrank K V, ¬LinearMap.ker (f ^ x) = LinearMap.ker (f ^ x.succ)
n : ℕ
ih : n ≤ (finrank K V).succ → n ≤ finrank K ↥(LinearMap.ker (f ^ n))
hn : n + 1 ≤ (finrank K V).succ
h_ker_lt_ker : LinearMap.ker (f ^ n) < LinearMap.ker (f ^ n.succ)
⊢ finrank K ↥(LinearMap.ker (f ^ n)) < finrank K ↥(LinearMap.ker (f ^ n.succ))
state_after: no goals