Module: Mathlib.GroupTheory.SchurZassenhaus

Goal state before:
pp: case refine_1
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : (map (QuotientGroup.mk' K) N).IsComplement' H
key : comap (QuotientGroup.mk' K) (map (QuotientGroup.mk' K) N) = N
⊢ comap (QuotientGroup.mk' K) (H ⊔ map (QuotientGroup.mk' K) N) = ⊤

pp: case refine_1.hf
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : (map (QuotientGroup.mk' K) N).IsComplement' H
key : comap (QuotientGroup.mk' K) (map (QuotientGroup.mk' K) N) = N
⊢ Function.Surjective ⇑(QuotientGroup.mk' K)

Tactic:
hH.symm.sup_eq_top,

Goal state after:
case refine_1
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : (map (QuotientGroup.mk' K) N).IsComplement' H
key : comap (QuotientGroup.mk' K) (map (QuotientGroup.mk' K) N) = N
⊢ comap (QuotientGroup.mk' K) ⊤ = ⊤

case refine_1.hf
G : Type u
inst✝³ : Group G
N : Subgroup G
inst✝² : N.Normal
h1 : (Nat.card ↥N).Coprime N.index
h2 :
  ∀ (G' : Type u) [inst : Group G'] [Finite G'],
    Nat.card G' < Nat.card G →
      ∀ {N' : Subgroup G'} [N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'
h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H
inst✝¹ : Finite G
K : Subgroup G
inst✝ : K.Normal
hK : K ≤ N
this : Function.Surjective ⇑(QuotientGroup.mk' K)
h4 : K ≠ ⊥ ∧ K ≠ N
h5 : Nat.card (G ⧸ K) < Nat.card G
h6 : (Nat.card ↥(map (QuotientGroup.mk' K) N)).Coprime (map (QuotientGroup.mk' K) N).index
H : Subgroup (G ⧸ K)
hH : (map (QuotientGroup.mk' K) N).IsComplement' H
key : comap (QuotientGroup.mk' K) (map (QuotientGroup.mk' K) N) = N
⊢ Function.Surjective ⇑(QuotientGroup.mk' K)