G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = ⊥ ∨ map (Subgroup.subtype N) K = N
⊢ K = ⊥ ∨ K = ⊤
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
⊢ K = ⊥ ∨ K = ⊤
have key := step2 h1 h2 h3 (K.map N.subtype) (map_subtype_le K)
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = N
⊢ K = ⊥ ∨ K = ⊤
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = ⊥ ∨ map (Subgroup.subtype N) K = N
⊢ K = ⊥ ∨ K = ⊤
rw [← map_bot N.subtype] at key
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊤
⊢ K = ⊥ ∨ K = ⊤
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = N
⊢ K = ⊥ ∨ K = ⊤
conv at key =>
  rhs
  rhs
  rw [← N.subtype_range, N.subtype.range_eq_map]
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊤
inj : Function.Injective (map (Subgroup.subtype N))
⊢ K = ⊥ ∨ K = ⊤
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊤
⊢ K = ⊥ ∨ K = ⊤
have inj := map_injective N.subtype_injective
no goals
G : Type u
inst✝³ : Group G
inst✝² : Fintype G
N : Subgroup G
inst✝¹ : Normal N
h1 : Nat.coprime (Fintype.card { x // x ∈ N }) (index N)
h2 :
  ∀ (G' : Type u) [inst : Group G'] [inst_1 : Fintype G'],
    Fintype.card G' < Fintype.card G →
      ∀ {N' : Subgroup G'} [inst_2 : Normal N'],
        Nat.coprime (Fintype.card { x // x ∈ N' }) (index N') → ∃ H', IsComplement' N' H'
h3 : ∀ (H : Subgroup G), ¬IsComplement' N H
K : Subgroup { x // x ∈ N }
inst✝ : Normal (map (Subgroup.subtype N) K)
key : map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊥ ∨ map (Subgroup.subtype N) K = map (Subgroup.subtype N) ⊤
inj : Function.Injective (map (Subgroup.subtype N))
⊢ K = ⊥ ∨ K = ⊤
rwa [inj.eq_iff, inj.eq_iff] at key