Theorem: OnCycleFactors.k_mul (Jordan/ConjClassCount.lean)

State before:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
⊢ k a σ ∘ k a τ = k a (σ * τ)
Tactic:
ext x
State after:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (k a σ ∘ k a τ) x = k a (σ * τ) x

State before:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ (k a σ ∘ k a τ) x = k a (σ * τ) x
Tactic:
simp only [Function.comp_apply]
State after:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ k a σ (k a τ x) = k a (σ * τ) x

State before:
case h
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
⊢ k a σ (k a τ x) = k a (σ * τ) x
Tactic:
by_cases hx : x ∈ g.support
State after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x

case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x

State before:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∈ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x
Tactic:
simp only [mem_support_iff_exists_Kf a] at hx
State after:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a σ (k a τ x) = k a (σ * τ) x

State before:
case pos
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : ∃ c i, x = Kf a 1 (c, i)
⊢ k a σ (k a τ x) = k a (σ * τ) x
Tactic:
obtain ⟨c, i, rfl⟩ := hx
State after:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ k a σ (k a τ (Kf a 1 (c, i))) = k a (σ * τ) (Kf a 1 (c, i))

State before:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ k a σ (k a τ (Kf a 1 (c, i))) = k a (σ * τ) (Kf a 1 (c, i))
Tactic:
rw [k_apply a c i hτ, k_apply_gen _]
State after:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a (σ * τ) (c, i) = k a (σ * τ) (Kf a 1 (c, i))

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card

State before:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a (σ * τ) (c, i) = k a (σ * τ) (Kf a 1 (c, i))

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
Tactic:
rw [k_apply_gen]
State after:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a (σ * τ) (c, i) = Kf a (σ * τ * 1) (c, i)

case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
    (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card

State before:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ Kf a (σ * τ) (c, i) = Kf a (σ * τ * 1) (c, i)

case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
    (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
Tactic:
simp only [mul_one]
State after:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
    (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card

α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card

State before:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
Tactic:
exact hσ
State after:
no goals

State before:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
⊢ ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }),
    (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card
Tactic:
intro c
State after:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card

State before:
case pos.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
c✝ : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
i : ℤ
c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
⊢ (Equiv.Perm.support ↑((σ * τ) c)).card = (Equiv.Perm.support ↑c).card
Tactic:
rw [Equiv.Perm.coe_mul, Function.comp_apply, hσ, hτ]
State after:
no goals

State before:
case neg
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Equiv.Perm α
a : Equiv.Perm.Basis g
σ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hσ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(σ c)).card = (Equiv.Perm.support ↑c).card
τ : Equiv.Perm { x // x ∈ Equiv.Perm.cycleFactorsFinset g }
hτ :
  ∀ (c : { x // x ∈ Equiv.Perm.cycleFactorsFinset g }), (Equiv.Perm.support ↑(τ c)).card = (Equiv.Perm.support ↑c).card
x : α
hx : x ∉ Equiv.Perm.support g
⊢ k a σ (k a τ x) = k a (σ * τ) x
Tactic:
simp only [k_apply_of_not_mem_support x hx]
State after:
no goals