-- Lean 4 proof step from Mathlib.RingTheory.Invariant.Profinite (goal)
Goal:
A : Type u_1
B : Type u_2
inst : CommRing A
inst_1 : CommRing B
inst_2 : Algebra A B
G : Type u
inst_3 : Group G
inst_4 : MulSemiringAction G B
inst_5 : SMulCommClass G A B
inst_6 : TopologicalSpace G
inst_7 : CompactSpace G
inst_8 : TotallyDisconnectedSpace G
inst_9 : IsTopologicalGroup G
inst_10 : TopologicalSpace B
inst_11 : DiscreteTopology B
inst_12 : ContinuousSMul G B
inst_13 : IsInvariant A B G
P Q : Ideal B
inst_14 : P.IsPrime
inst_15 : Q.IsPrime
hPQ : Eq (α := Ideal A) (Ideal.under A P) (Ideal.under A Q)
B' : (G : Type u) → [inst_16 : Monoid G] → [inst_17 : MulSemiringAction G B] → [SMulCommClass G A B] → Subalgebra A B :=
  FixedPoints.subalgebra A B
N : OpenNormalSubgroup G
⊢ (↾fun
        (x :
          { g : G ⧸ (↑N.toOpenSubgroup : Subgroup G) //
            Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
              HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P) }) ↦
      Subtype.mk (p := fun (g : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
        Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
          HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
            (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P))
        ((QuotientGroup.map (↑N.toOpenSubgroup : Subgroup G) N.toOpenSubgroup.1 (MonoidHom.id G) ⋯ :
            (a : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) → G ⧸ N.toOpenSubgroup.1)
          (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
        ⋯) =
    𝟙
      { g : G ⧸ (↑N.toOpenSubgroup : Subgroup G) //
        Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
          HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
            (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P) }
have h : ∀ (val : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (x : G)
  (hx :
    Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
      HSMul.hSMul (α := G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G)))
        (Quot.mk (⇑(QuotientGroup.leftRel (↑N.toOpenSubgroup : Subgroup G)) : G → G → Prop) x)
        (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P)),
  (↑(TypeCat.Fun.toFun (X :=
          { g : G ⧸ (↑N.toOpenSubgroup : Subgroup G) //
            Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
              HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P) })
          (ConcreteCategory.hom (FC := TypeCat.Fun)
            (↾fun
                (x :
                  { g : G ⧸ (↑N.toOpenSubgroup : Subgroup G) //
                    Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
                      HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                        (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P) }) ↦
              Subtype.mk (p := fun (g : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
                Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
                  HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                    (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P))
                ((QuotientGroup.map (↑N.toOpenSubgroup : Subgroup G) N.toOpenSubgroup.1 (MonoidHom.id G) sorry :
                    (a : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) → G ⧸ N.toOpenSubgroup.1)
                  (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
                sorry))
          (Subtype.mk (p := fun (g : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
            Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
              HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P))
            (Quot.mk (⇑(QuotientGroup.leftRel (↑N.toOpenSubgroup : Subgroup G)) : G → G → Prop) x) hx)) :
      G ⧸ (↑N.toOpenSubgroup : Subgroup G)) =
    (↑((ConcreteCategory.hom (FC := TypeCat.Fun)
              (𝟙
                { g : G ⧸ (↑N.toOpenSubgroup : Subgroup G) //
                  Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
                    HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                      (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P) })).toFun
          (Subtype.mk (p := fun (g : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
            Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) Q =
              HSMul.hSMul (β := Ideal ↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) g
                (Ideal.under (↥(B' ↥(↑N.toOpenSubgroup : Subgroup G))) P))
            (Quot.mk (⇑(QuotientGroup.leftRel (↑N.toOpenSubgroup : Subgroup G)) : G → G → Prop) x) hx)) :
      G ⧸ (↑N.toOpenSubgroup : Subgroup G))
Tactic: ext ⟨⟨x⟩, hx⟩