-- Mathlib/GroupTheory/Perm/Centralizer.lean
lemma Subgroup.Centralizer.toConjAct_smul_mem_cycleFactorsFinset {k c : Perm α}
    (k_mem : k ∈ centralizer {g}) (c_mem : c ∈ g.cycleFactorsFinset) :
    ConjAct.toConjAct k • c ∈ g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : (ConjAct.toConjAct k • g).cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
rw [ConjAct.toConjAct_smul, mem_centralizer_singleton_iff.mp k_mem, mul_assoc] at this

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : (g * (k * k⁻¹)).cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset
⊢ ConjAct.toConjAct k • c ∈ ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
simp only [Set.smul_mem_smul_set_iff, Finset.mem_coe, Finset.coe_mem, c_mem]

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
have this := cycleFactorsFinset_conj_eq (ConjAct.toConjAct (k : Perm α)) g

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : (ConjAct.toConjAct k • g).cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
⊢ ConjAct.toConjAct k • c ∈ g.cycleFactorsFinset

Tactic:
suffices (g.cycleFactorsFinset : Set (Perm α)) =
  (ConjAct.toConjAct k) • g.cycleFactorsFinset by
  rw [← Finset.mem_coe, this]
  simp only [Set.smul_mem_smul_set_iff, Finset.mem_coe, Finset.coe_mem, c_mem]

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset
⊢ ConjAct.toConjAct k • c ∈ g.cycleFactorsFinset

Tactic:
rw [← Finset.mem_coe, this]

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset
⊢ ConjAct.toConjAct k • c ∈ ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : g.cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
conv_lhs => rw [this]

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : g.cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑(ConjAct.toConjAct k • g.cycleFactorsFinset) = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : g.cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑(ConjAct.toConjAct k • g.cycleFactorsFinset) = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
simp only [Finset.coe_smul_finset]

Result:
no goals

Proof state:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : (g * (k * k⁻¹)).cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset

Tactic:
simp only [mul_inv_cancel, mul_one] at this

Result:
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g k c : Perm α
k_mem : k ∈ centralizer {g}
c_mem : c ∈ g.cycleFactorsFinset
this : g.cycleFactorsFinset = ConjAct.toConjAct k • g.cycleFactorsFinset
⊢ ↑g.cycleFactorsFinset = ConjAct.toConjAct k • ↑g.cycleFactorsFinset