-- Lean 4 proof step from Mathlib.RingTheory.Invariant.Basic (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_3
inst_3 : Group G
inst_4 : Finite G
inst_5 : MulSemiringAction G B
inst_6 : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst_7 : Q.IsPrime
inst_8 : Q.LiesOver P
K : Type u_4
L : Type u_5
inst_9 : Field K
inst_10 : Field L
inst_11 : Algebra.{u_1, u_4} (A ⧸ P) K
inst_12 : Algebra.{u_2, u_5} (B ⧸ Q) L
inst_13 : Algebra.{u_1, u_5} (A ⧸ P) L
inst_14 : IsScalarTower.{u_1, u_2, u_5} (A ⧸ P) (B ⧸ Q) L
inst_15 : Algebra K L
inst_16 : IsScalarTower.{u_1, u_4, u_5} (A ⧸ P) K L
inst_17 : Algebra.IsInvariant A B G
inst_18 : Module.IsTorsionFree.{u_2, u_5} (B ⧸ Q) L
f : Gal(L/K)
b : B ⧸ Q
hx :
  ∀ (g : ↥(MulAction.stabilizer G Q)),
    ((((Ideal.Quotient.stabilizerHom Q P G :
                  MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                    (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
              g :
            AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
          (a : B ⧸ Q) → B ⧸ Q)
        b =
      b
intro :
  ∀ (val : Fintype G),
    Eq (α := L) ((f : (a : L) → L) ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) b : L) : L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) b : L)
intro₁ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q)),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) : L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) : L)
intro₂ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q)),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
              (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
            L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
        L)
intro₃ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q))
    (a b : B) (ha1 : ∀ (g : G), g • a = a) (ha2 : a ∉ Q)
    (hb :
      ∀ (g : G),
        Eq (α := B ⧸ Q) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (g • b) : B ⧸ Q)
          ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (if g • Q = Q then a * b₀ else 0) : B ⧸ Q)),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
              (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
            L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
        L)
intro₄ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q))
    (a b : B) (ha1 : ∀ (g : G), g • a = a) (ha2 : a ∉ Q)
    (hb :
      ∀ (g : G),
        Eq (α := B ⧸ Q) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (g • b) : B ⧸ Q)
          ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (if g • Q = Q then a * b₀ else 0) : B ⧸ Q))
    (M : A[X]) (key : Eq (α := B[X]) (map (algebraMap A B) M) (MulSemiringAction.charpoly G b)),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
              (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
            L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
        L)
intro₅ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q))
    (a b : B) (ha1 : ∀ (g : G), g • a = a) (ha2 : a ∉ Q)
    (hb :
      ∀ (g : G),
        Eq (α := B ⧸ Q) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (g • b) : B ⧸ Q)
          ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (if g • Q = Q then a * b₀ else 0) : B ⧸ Q))
    (M : A[X])
    (key :
      Eq (α := (B ⧸ Q)[X]) (map (algebraMap B (B ⧸ Q)) (map (algebraMap A B) M))
        (map (algebraMap B (B ⧸ Q)) (MulSemiringAction.charpoly G b))),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
              (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
            L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
        L)
intro₆ :
  ∀ (val : Fintype G) (b₀ : B)
    (hx :
      ∀ (g : ↥(MulAction.stabilizer G Q)),
        Eq (α := B ⧸ Q)
          (((((Ideal.Quotient.stabilizerHom Q P G :
                        MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                          (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                      (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                    g :
                  AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
                (a : B ⧸ Q) → B ⧸ Q)
              ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
            B ⧸ Q)
          ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q))
    (a b : B) (ha1 : ∀ (g : G), g • a = a) (ha2 : a ∉ Q)
    (hb :
      ∀ (g : G),
        Eq (α := B ⧸ Q) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (g • b) : B ⧸ Q)
          ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (if g • Q = Q then a * b₀ else 0) : B ⧸ Q))
    (M : A[X])
    (key :
      Eq.{u_2 + 1} (α := (B ⧸ Q)[X]) (map (R := A ⧸ P) (algebraMap (A ⧸ P) (B ⧸ Q)) (map (algebraMap A (A ⧸ P)) M))
        (∏ i : G,
          map (algebraMap B (B ⧸ Q)) (HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) (i • b) : B[X])))),
    Eq (α := L)
      ((f : (a : L) → L)
          ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
              (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
            L) :
        L)
      ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
        L)
⊢ Eq (α := L) ((f : (a : L) → L) ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) b : L) : L)
    ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) b : L)
have intro₇ : ∀ (val : Fintype G) (b₀ : B)
  (hx :
    ∀ (g : ↥(MulAction.stabilizer G Q)),
      Eq (α := B ⧸ Q)
        (((((Ideal.Quotient.stabilizerHom Q P G :
                      MonoidHom.{u_3, u_2} (↥(MulAction.stabilizer G Q))
                        (AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q))) :
                    (a : ↥(MulAction.stabilizer G Q)) → (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q)
                  g :
                AlgEquiv.{u_1, u_2, u_2} (A ⧸ P) (B ⧸ Q) (B ⧸ Q)) :
              (a : B ⧸ Q) → B ⧸ Q)
            ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀) :
          B ⧸ Q)
        ((Ideal.Quotient.mk Q : (a : B) → B ⧸ Q) b₀ : B ⧸ Q))
  (a b : B) (ha1 : ∀ (g : G), g • a = a) (ha2 : a ∉ Q)
  (hb :
    ∀ (g : G),
      Eq (α := B ⧸ Q) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (g • b) : B ⧸ Q)
        ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (if g • Q = Q then a * b₀ else 0) : B ⧸ Q))
  (M : A[X])
  (key :
    Eq.{u_2 + 1} (α := (B ⧸ Q)[X]) (map (R := A ⧸ P) (algebraMap (A ⧸ P) (B ⧸ Q)) (map (algebraMap A (A ⧸ P)) M))
      (∏ i : G, map (algebraMap B (B ⧸ Q)) (HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) (i • b) : B[X]))))
  (key₀ :
    ∀ (g : G),
      map (algebraMap B (B ⧸ Q)) (HSub.hSub (α := B[X]) (β := B[X]) X ((C : (a : B) → B[X]) (g • b) : B[X])) =
        if g • Q = Q then
          HSub.hSub (α := (B ⧸ Q)[X]) (β := (B ⧸ Q)[X]) X
            ((C : (a : B ⧸ Q) → (B ⧸ Q)[X]) ((algebraMap B (B ⧸ Q) : (a : B) → B ⧸ Q) (a * b₀)) : (B ⧸ Q)[X])
        else X),
  Eq (α := L)
    ((f : (a : L) → L)
        ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L)
            (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) :
          L) :
      L)
    ((algebraMap (B ⧸ Q) L : (a : B ⧸ Q) → L) (((algebraMap B (B ⧸ Q) : B →+* B ⧸ Q) : (a : B) → B ⧸ Q) b₀ : B ⧸ Q) : L)
Tactic: classical
  cases nonempty_fintype G
  obtain ⟨b₀, rfl⟩ := Ideal.Quotient.mk_surjective b
  rw [← Ideal.Quotient.algebraMap_eq]
  obtain ⟨a, b, ha1, ha2, hb⟩ := fixed_of_fixed1_aux2 G Q b₀ (fun g hg ↦ hx ⟨g, hg⟩)
  obtain ⟨M, key⟩ := (mem_lifts _).mp (Algebra.IsInvariant.charpoly_mem_lifts A B G b)
  replace key := congrArg (map (algebraMap B (B ⧸ Q))) key
  rw [map_map, ← algebraMap_eq, algebraMap_eq A (A ⧸ P) (B ⧸ Q),
      ← map_map, MulSemiringAction.charpoly, Polynomial.map_prod] at key
  have key₀ : ∀ g : G, (X - C (g • b)).map (algebraMap B (B ⧸ Q)) =
      if g • Q = Q then X - C (algebraMap B (B ⧸ Q) (a * b₀)) else X := by
    intro g
    rw [Polynomial.map_sub, map_X, map_C, hb]
    split_ifs
    · rfl
    · rw [map_zero, map_zero, sub_zero]
  simp only [key₀, Finset.prod_ite, Finset.prod_const] at key
  replace key := congrArg (map (algebraMap (B ⧸ Q) L)) key
  rw [map_map, ← algebraMap_eq, algebraMap_eq (A ⧸ P) K L,
      ← map_map, Polynomial.map_mul, Polynomial.map_pow, Polynomial.map_pow, Polynomial.map_sub,
      map_X, map_C] at key
  replace key := fixed_of_fixed1_aux3 key f (Finset.card_ne_zero_of_mem
    (Finset.mem_filter.mpr ⟨Finset.mem_univ 1, one_smul G Q⟩))
  simp only [map_mul] at key
  obtain ⟨a, rfl⟩ := Algebra.IsInvariant.isInvariant (A := A) a ha1
  rwa [← algebraMap_apply A B (B ⧸ Q), algebraMap_apply A (A ⧸ P) (B ⧸ Q),
      ← algebraMap_apply, algebraMap_apply (A ⧸ P) K L, f.commutes, mul_right_inj'] at key
  rwa [← algebraMap_apply, algebraMap_apply (A ⧸ P) (B ⧸ Q) L,
      ← algebraMap_apply A (A ⧸ P) (B ⧸ Q), algebraMap_apply A B (B ⧸ Q),
      Ne, algebraMap_eq_zero_iff, Ideal.Quotient.algebraMap_eq, Ideal.Quotient.eq_zero_iff_mem]
Cleared hypotheses: b, hx