Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)

Tactic:
by
  by_cases hM : 0 ∈ M
  · subsingleton [IsLocalization.uniqueOfZeroMem (S := Aₘ) hM]
  replace hM : M ≤ A⁰ := fun x hx ↦ mem_nonZeroDivisors_iff_ne_zero.mpr (fun e ↦ hM (e ▸ hx))
  let K := FractionRing A
  let L := FractionRing B
  have : IsIntegralClosure B A L :=
    IsIntegralClosure.of_isIntegrallyClosed _ _ _
  -- TODO: How is this even supposed to fire? `R` and `S` cannot be inferred.
  have : Algebra.IsAlgebraic (FractionRing A) (FractionRing B) :=
    isAlgebraic_of_isFractionRing A B ..
  have : IsLocalization (algebraMapSubmonoid B A⁰) L :=
    IsIntegralClosure.isLocalization _ (FractionRing A) _ _
  let f : Aₘ →+* K := IsLocalization.map _ (T := A⁰) (RingHom.id A) hM
  let := f.toAlgebra
  have : IsScalarTower A Aₘ K := IsScalarTower.of_algebraMap_eq'
    (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHomCompTriple.comp_eq])
  let := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
  let g : Bₘ →+* L := IsLocalization.map _
      (M := algebraMapSubmonoid B M) (T := algebraMapSubmonoid B A⁰)
      (RingHom.id B) (Submonoid.monotone_map hM)
  let := g.toAlgebra
  have : IsScalarTower B Bₘ L := IsScalarTower.of_algebraMap_eq'
    (by rw [RingHom.algebraMap_toAlgebra, IsLocalization.map_comp, RingHomCompTriple.comp_eq])
  let := ((algebraMap K L).comp f).toAlgebra
  have : IsScalarTower Aₘ K L := IsScalarTower.of_algebraMap_eq' rfl
  have : IsScalarTower Aₘ Bₘ L := by
    apply IsScalarTower.of_algebraMap_eq'
    apply IsLocalization.ringHom_ext M
    rw [RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra (R := Bₘ), RingHom.comp_assoc,
      RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq, IsScalarTower.algebraMap_eq A B Bₘ,
      IsLocalization.map_comp, RingHom.comp_id, ← RingHom.comp_assoc, IsLocalization.map_comp,
      RingHom.comp_id, ← IsScalarTower.algebraMap_eq, ← IsScalarTower.algebraMap_eq]
  let := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization
    (algebraMapSubmonoid B M) Bₘ L
  have : FiniteDimensional K L := .of_isLocalization A B A⁰
  have : IsIntegralClosure Bₘ Aₘ L :=
    IsIntegralClosure.of_isIntegrallyClosed _ _ _
  apply IsFractionRing.injective Aₘ K
  rw [← IsScalarTower.algebraMap_apply, Algebra.algebraMap_intTrace_fractionRing,
    Algebra.algebraMap_intTrace (L := L), ← IsScalarTower.algebraMap_apply]

Goal state after: