-- Module: Mathlib.RingTheory.Etale.QuasiFinite
Goal:
R : Type u
S : Type v
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : FiniteType R S
p : Ideal R
inst_4 : p.IsPrime
q : Ideal S
inst_5 : q.IsPrime
inst_6 : q.LiesOver p
inst_7 : QuasiFiniteAt R q
R' : Type u
w : CommRing R'
w_1 : Algebra R R'
w_2 : Etale R R'
P : Ideal R'
w_3 : P.IsPrime
w_4 : P.LiesOver p
e : R' ⊗[R] S
he : IsIdempotentElem e
e₀ : TensorProduct.{u, u, v} R R' ↥(integralClosure R S)
he₀ : IsIdempotentElem e₀
he₀e :
  ((TensorProduct.map (AlgHom.id R' R') (Subalgebra.val (integralClosure R S)) :
          TensorProduct.{u, u, v} R R' ↥(integralClosure R S) →ₐ[R'] R' ⊗[R] S) :
        (a : R' ⊗[R] ↥(integralClosure R S)) → R' ⊗[R] S)
      e₀ =
    e
P' : Ideal (R' ⊗[R] S)
w_5 : P'.IsPrime
w_6 : P'.LiesOver P
hP'q : Ideal.comap (F := S →+* R' ⊗[R] S) (AlgHom.toRingHom TensorProduct.includeRight) P' = q
heP' : e ∉ P'
hpP :
  Function.Bijective (α := p.ResidueField) (β := P.ResidueField)
    (⇑(Ideal.ResidueField.mapₐ p P (ofId R R') ⋯) : (a : p.ResidueField) → P.ResidueField)
H' :
  ∀ (P'' : Ideal (TensorProduct.{u, u, v} R R' ↥(integralClosure R S))),
    P''.IsPrime →
      P''.LiesOver P →
        e₀ ∉ P'' →
          P'' =
            Ideal.comap (F := TensorProduct.{u, u, v} R R' ↥(integralClosure R S) →+* R' ⊗[R] S)
              (AlgHom.toRingHom
                (TensorProduct.map (AlgHom.id R' R') (Subalgebra.val (integralClosure R S)) :
                  TensorProduct.{u, u, v} R R' ↥(integralClosure R S) →ₐ[R'] R' ⊗[R] S))
              P'
H : ∀ (P'' : Ideal (R' ⊗[R] S)), P''.IsPrime → P''.LiesOver P → e ∉ P'' → P'' = P'
g : ↥(integralClosure R S)
hgq : (↑g : S) ∉ q
hg :
  Function.Bijective (α := Localization.Away g) (β :=
    Localization.Away (M := S)
      ((AlgHom.toRingHom (Subalgebra.val (integralClosure R S)) : (a : ↥(integralClosure R S)) → S) g : S))
    (⇑(Localization.awayMap (AlgHom.toRingHom (Subalgebra.val (integralClosure R S))) g) :
      (a : Localization.Away g) →
        Localization.Away (((integralClosure R S).val.toRingHom : (a : ↥(integralClosure R S)) → S) g))
f : R'
hfP : f ∉ P
hf :
  Module.Finite (Localization.Away f)
    (Localization.Away (M := Localization.Away f ⊗[R] S)
      (((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S) :
              R' ⊗[R] S →ₐ[R'] Localization.Away f ⊗[R] S) :
            (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S)
          e :
        Localization.Away f ⊗[R] S))
Pf : Ideal (Localization.Away f) := Ideal.map (F := R' →+* Localization.Away f) (algebraMap R' (Localization.Away f)) P
this_1 : Pf.IsPrime
this_2 : Pf.LiesOver P
φ : R' ⊗[R] S →ₐ[R'] Localization.Away f ⊗[R] S := TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)
this_3 : Algebra (R' ⊗[R] S) (Localization.Away f ⊗[R] S) := φ.toAlgebra
this_4 : IsScalarTower R' (R' ⊗[R] S) (Localization.Away f ⊗[R] S) := ⋯
this_5 : IsLocalization.Away (R := R' ⊗[R] S) (f ⊗ₜ[R] 1) (Localization.Away f ⊗[R] S)
P'f : Ideal (Localization.Away f ⊗[R] S) :=
  Ideal.map (F := R' ⊗[R] S →+* Localization.Away f ⊗[R] S) (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S)) P'
hP'f : Disjoint (↑(Submonoid.powers (f ⊗ₜ[R] 1)) : Set (R' ⊗[R] S)) (↑P' : Set (R' ⊗[R] S))
this_6 : P'f.IsPrime
this_7 : P'f.LiesOver (A := R' ⊗[R] S) P'
this_8 : P'f.LiesOver P
this_9 : P'f.LiesOver Pf
P'' : Ideal (Localization.Away f ⊗[R] S)
h : P''.IsPrime
h_1 : P''.LiesOver Pf
hP'' :
  (((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S) : R' ⊗[R] S →ₐ[R'] Localization.Away f ⊗[R] S) :
          (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S)
        e :
      Localization.Away f ⊗[R] S) ∉
    P''
this : P''.LiesOver P
⊢ Eq (α := Ideal (R' ⊗[R] S))
    (PrimeSpectrum.asIdeal
      (PrimeSpectrum.comap (S := Localization.Away f ⊗[R] S) (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S))
        { asIdeal := P'', isPrime := h }))
    (PrimeSpectrum.asIdeal
      (PrimeSpectrum.comap (S := Localization.Away f ⊗[R] S) (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S))
        { asIdeal := P'f, isPrime := this_6 }))

Tactic:
exact (H (P''.under _) inferInstance inferInstance hP'').trans (P'f.over_def P')

Premises used: inferInstance, Ideal.over_def, Eq.trans, Ideal.under