Theorem: Equiv.Perm.sign_subtypePerm
File: Mathlib/GroupTheory/Perm/Sign.lean

Proof steps:
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
⊢ sign (f.subtypePerm h₁) = sign f
state_after: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
⊢ sign (f.subtypePerm h₁) = sign f
tactic: let l := (truncSwapFactors (subtypePerm f h₁)).out
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
⊢ sign (f.subtypePerm h₁) = sign f
state_after: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
⊢ sign (f.subtypePerm h₁) = sign f
tactic: have hl' : ∀ g' ∈ l.1.map ofSubtype, IsSwap g' := fun g' hg' =>
    let ⟨g, hg⟩ := List.mem_map.1 hg'
    hg.2 ▸ (l.2.2 _ hg.1).of_subtype_isSwap
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
⊢ sign (f.subtypePerm h₁) = sign f
state_after: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (f.subtypePerm h₁) = sign f
tactic: have hl'₂ : (l.1.map ofSubtype).prod = f := by
    rw [l.1.prod_hom ofSubtype, l.2.1, ofSubtype_subtypePerm _ h₂]
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (f.subtypePerm h₁) = sign f
state_after: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (↑l).prod = sign f
tactic: conv =>
    congr
    rw [← l.2.1]
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (↑l).prod = sign f
state_after: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (↑l).prod = sign (List.map ⇑ofSubtype ↑l).prod
tactic: simp_rw [← hl'₂]
state_before: α : Type u
inst✝² : DecidableEq α
β : Type v
inst✝¹ : Fintype α
f : Perm α
p : α → Prop
inst✝ : DecidablePred p
h₁ : ∀ (x : α), p x ↔ p (f x)
h₂ : ∀ (x : α), f x ≠ x → p x
l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out
hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap
hl'₂ : (List.map ⇑ofSubtype ↑l).prod = f
⊢ sign (↑l).prod = sign (List.map ⇑ofSubtype ↑l).prod
state_after: no goals
tactic: rw [sign_prod_list_swap l.2.2, sign_prod_list_swap hl', List.length_map]