-- 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'
x :
  { 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' := ⋯ })
        (↑((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' := ⋯ })) }
⊢ 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))

Tactic:
have h : FixedPoints.subalgebra A B N'.1.1 ≤ FixedPoints.subalgebra A B N.1.1 :=
      fun x hx n ↦ hx ⟨_, i.le n.2⟩