-- Module: Mathlib.RingTheory.Invariant.Profinite
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
P_1 : Ideal A
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
P : Ideal A
Q : Ideal B
inst_13 : Q.LiesOver P
σ : AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)
N N' : OpenNormalSubgroup G
i : N ⟶ N'
h :
  LE.le (α := Subalgebra A B) (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))
    (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))
x : G
hx :
  Membership.mem (γ := Subgroup (G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
    (MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
      Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
      (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
    (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G))
hx' :
  Eq (α :=
    AlgHom.{u_1, u_2, u_2} (A ⧸ P)
      (HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
        Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
      (B ⧸ Q))
    (AlgHom.comp (B :=
      HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
        Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
      (let __RingHom :
        HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
            Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
            (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q) →+*
          B ⧸ Q :=
        quotientMap Q
          (Subsemiring.subtype
            (Subalgebra.toSubsemiring (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))))
          ⋯;
      { toRingHom := __RingHom, commutes' := ⋯ })
      (↑((stabilizerHom (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q) P
                (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) :
              (a :
                  ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G))
                      (under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))) →
                (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q) ≃ₐ[A ⧸ P]
                  ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q)
            (Subtype.mk (p := fun (x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
              Membership.mem (γ := Subgroup (G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
                (MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
                  Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                  (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
                x)
              (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) hx)) :
        ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q →ₐ[A ⧸ P]
          ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
    ((↑σ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q).comp
      (let __RingHom :
        HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
            Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
            (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q) →+*
          B ⧸ Q :=
        quotientMap Q
          (Subsemiring.subtype
            (Subalgebra.toSubsemiring (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))))
          ⋯;
      { toRingHom := __RingHom, commutes' := ⋯ }))
H :
  ∀
    (g :
      HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
        Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q)),
    Eq (α := B ⧸ Q)
      ((AlgHom.comp (B :=
              HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
                Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
                (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
              (let __RingHom :
                HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))
                    (B := Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
                    (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →+*
                  B ⧸ Q :=
                quotientMap Q
                  (Subsemiring.subtype
                    (Subalgebra.toSubsemiring
                      (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))))
                  ⋯;
              { toRingHom := __RingHom, commutes' := ⋯ })
              (↑((stabilizerHom
                        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) P
                        (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) :
                      (a :
                          ↥(MulAction.stabilizer (G ⧸ (↑N'.toOpenSubgroup : Subgroup G))
                              (under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))) →
                        (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                            under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) ≃ₐ[A ⧸ P]
                          ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                            under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q)
                    (Subtype.mk (p := fun (x : G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) ↦
                      Membership.mem (γ := Subgroup (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)))
                        (MulAction.stabilizer (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) (α :=
                          Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
                          (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
                        x)
                      ((QuotientGroup.map (↑N.toOpenSubgroup : Subgroup G) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯ :
                          (a : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) → G ⧸ N'.toOpenSubgroup.1)
                        (↑(↑(Subtype.mk (p :=
                                  fun
                                    (σ' :
                                      ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
                                          Ideal
                                            ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G)))
                                          (under
                                            (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G)))
                                            Q))) ↦
                                  Eq (α :=
                                    AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                                      (HasQuotient.Quotient
                                        ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))
                                        (B :=
                                        Ideal
                                          ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                        (under
                                          (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                              ↥(↑N.toOpenSubgroup : Subgroup G)))
                                          Q))
                                      (B ⧸ Q))
                                    (AlgHom.comp (B :=
                                      HasQuotient.Quotient
                                        ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))
                                        (B :=
                                        Ideal
                                          ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                        (under
                                          (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                              ↥(↑N.toOpenSubgroup : Subgroup G)))
                                          Q))
                                      (let __RingHom :
                                        HasQuotient.Quotient
                                            ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))
                                            (B :=
                                            Ideal
                                              ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                            (under
                                              (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q) →+*
                                          B ⧸ Q :=
                                        quotientMap Q
                                          (Subsemiring.subtype
                                            (Subalgebra.toSubsemiring
                                              (FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))))
                                          ⋯;
                                      { toRingHom := __RingHom, commutes' := ⋯ :
                                        AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                                          (HasQuotient.Quotient
                                            ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))
                                            (B :=
                                            Ideal
                                              ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                            (under
                                              (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q))
                                          (B ⧸ Q) })
                                      (↑((stabilizerHom
                                                (under
                                                  (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                      ↥(↑N.toOpenSubgroup : Subgroup G)))
                                                  Q)
                                                P (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) :
                                              (a :
                                                  ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G))
                                                      (under
                                                        (↥(FixedPoints.subalgebra A B
                                                            ↥(↑N.toOpenSubgroup : Subgroup G)))
                                                        Q))) →
                                                (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                                    under
                                                      (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                                      Q) ≃ₐ[A ⧸ P]
                                                  ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                                    under
                                                      (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                                      Q)
                                            σ') :
                                        ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q →ₐ[A ⧸ P]
                                          ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
                                    ((↑σ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q).comp
                                      (let __RingHom :
                                        HasQuotient.Quotient
                                            ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))
                                            (B :=
                                            Ideal
                                              ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                            (under
                                              (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q) →+*
                                          B ⧸ Q :=
                                        quotientMap Q
                                          (Subsemiring.subtype
                                            (Subalgebra.toSubsemiring
                                              (FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))))
                                          ⋯;
                                      { toRingHom := __RingHom, commutes' := ⋯ :
                                        AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                                          (HasQuotient.Quotient
                                            ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                ↥(↑N.toOpenSubgroup : Subgroup G))
                                            (B :=
                                            Ideal
                                              ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                            (under
                                              (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                                  ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q))
                                          (B ⧸ Q) })))
                                  (Subtype.mk (p := fun (x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
                                    Membership.mem (γ := Subgroup (G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
                                      (MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
                                        Ideal
                                          ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                        (under
                                          (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                              ↥(↑N.toOpenSubgroup : Subgroup G)))
                                          Q))
                                      x)
                                    (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) hx)
                                  hx') :
                              ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G))
                                  (under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))) :
                          G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
                      ⋯)) :
                ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q →ₐ[A ⧸ P]
                  ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) :
            (a :
                ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                  under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →
              B ⧸ Q)
          g :
        B ⧸ Q)
      (((↑σ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q).comp
              (let __RingHom :
                HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))
                    (B := Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
                    (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →+*
                  B ⧸ Q :=
                quotientMap Q
                  (Subsemiring.subtype
                    (Subalgebra.toSubsemiring
                      (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))))
                  ⋯;
              { toRingHom := __RingHom, commutes' := ⋯ }) :
            (a :
                ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                  under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →
              B ⧸ Q)
          g :
        B ⧸ Q)
⊢ Eq (α :=
    AlgHom.{u_1, u_2, u_2} (A ⧸ P)
      (HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
        Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
      (B ⧸ Q))
    (AlgHom.comp (B :=
      HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
        Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
        (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
      (let __RingHom :
        HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
            Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
            (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →+*
          B ⧸ Q :=
        quotientMap Q
          (Subsemiring.subtype
            (Subalgebra.toSubsemiring (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))))
          ⋯;
      { toRingHom := __RingHom, commutes' := ⋯ })
      (↑((stabilizerHom (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) P
                (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) :
              (a :
                  ↥(MulAction.stabilizer (G ⧸ (↑N'.toOpenSubgroup : Subgroup G))
                      (under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))) →
                (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) ≃ₐ[A ⧸ P]
                  ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
                    under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q)
            (Subtype.mk (p := fun (x : G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) ↦
              Membership.mem (γ := Subgroup (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)))
                (MulAction.stabilizer (G ⧸ (↑N'.toOpenSubgroup : Subgroup G)) (α :=
                  Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
                  (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
                x)
              ((QuotientGroup.map (↑N.toOpenSubgroup : Subgroup G) N'.toOpenSubgroup.1 (MonoidHom.id G) ⋯ :
                  (a : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) → G ⧸ N'.toOpenSubgroup.1)
                (↑(↑(Subtype.mk (p :=
                          fun
                            (σ' :
                              ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
                                  Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                  (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                    Q))) ↦
                          Eq (α :=
                            AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                              (HasQuotient.Quotient
                                ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                  Q))
                              (B ⧸ Q))
                            (AlgHom.comp (B :=
                              HasQuotient.Quotient
                                ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                  Q))
                              (let __RingHom :
                                HasQuotient.Quotient
                                    ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                    Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                    (under
                                      (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                      Q) →+*
                                  B ⧸ Q :=
                                quotientMap Q
                                  (Subsemiring.subtype
                                    (Subalgebra.toSubsemiring
                                      (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))))
                                  ⋯;
                              { toRingHom := __RingHom, commutes' := ⋯ :
                                AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                                  (HasQuotient.Quotient
                                    ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                    Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                    (under
                                      (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                      Q))
                                  (B ⧸ Q) })
                              (↑((stabilizerHom
                                        (under
                                          (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B
                                              ↥(↑N.toOpenSubgroup : Subgroup G)))
                                          Q)
                                        P (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) :
                                      (a :
                                          ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G))
                                              (under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                                Q))) →
                                        (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                              Q) ≃ₐ[A ⧸ P]
                                          ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                            under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q)
                                    σ') :
                                ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                    under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q →ₐ[A ⧸ P]
                                  ↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G)) ⧸
                                    under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))
                            ((↑σ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q).comp
                              (let __RingHom :
                                HasQuotient.Quotient
                                    ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                    Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                    (under
                                      (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                      Q) →+*
                                  B ⧸ Q :=
                                quotientMap Q
                                  (Subsemiring.subtype
                                    (Subalgebra.toSubsemiring
                                      (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G))))
                                  ⋯;
                              { toRingHom := __RingHom, commutes' := ⋯ :
                                AlgHom.{u_1, u_2, u_2} (A ⧸ P)
                                  (HasQuotient.Quotient
                                    ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)) (B :=
                                    Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                    (under
                                      (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                      Q))
                                  (B ⧸ Q) })))
                          (Subtype.mk (p := fun (x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) ↦
                            Membership.mem (γ := Subgroup (G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
                              (MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G)) (α :=
                                Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N.toOpenSubgroup : Subgroup G)))
                                  Q))
                              x)
                            (↑x : G ⧸ (↑N.toOpenSubgroup : Subgroup G)) hx)
                          hx') :
                      ↥(MulAction.stabilizer (G ⧸ (↑N.toOpenSubgroup : Subgroup G))
                          (under (↥(FixedPoints.subalgebra A B ↥(↑N.toOpenSubgroup : Subgroup G))) Q))) :
                  G ⧸ (↑N.toOpenSubgroup : Subgroup G)))
              ⋯)) :
        ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
            under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q →ₐ[A ⧸ P]
          ↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G)) ⧸
            under (↥(FixedPoints.subalgebra A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q))
    ((↑σ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q).comp
      (let __RingHom :
        HasQuotient.Quotient ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)) (B :=
            Ideal ↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G)))
            (under (↥(FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))) Q) →+*
          B ⧸ Q :=
        quotientMap Q
          (Subsemiring.subtype
            (Subalgebra.toSubsemiring (FixedPoints.subalgebra.{u_1, u_2, u} A B ↥(↑N'.toOpenSubgroup : Subgroup G))))
          ⋯;
      { toRingHom := __RingHom, commutes' := ⋯ }))

Tactic:
ext g