Module: Mathlib.RingTheory.Etale.QuasiFinite

Goal state before:
pp: case refine_4
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FiniteType R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
inst✝¹ : q.LiesOver p
inst✝ : QuasiFiniteAt R q
R' : Type u
w✝⁶ : CommRing R'
w✝⁵ : Algebra R R'
w✝⁴ : Etale R R'
P : Ideal R'
w✝³ : P.IsPrime
w✝² : P.LiesOver p
e : R' ⊗[R] S
he : IsIdempotentElem e
e₀ : R' ⊗[R] ↥(integralClosure R S)
he₀ : IsIdempotentElem e₀
he₀e : (TensorProduct.map (AlgHom.id R' R') (integralClosure R S).val) e₀ = e
P' : Ideal (R' ⊗[R] S)
w✝¹ : P'.IsPrime
w✝ : P'.LiesOver P
hP'q : Ideal.comap TensorProduct.includeRight.toRingHom P' = q
heP' : e ∉ P'
hpP : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p P (ofId R R') ⋯)
H' :
  ∀ (P'' : Ideal (R' ⊗[R] ↥(integralClosure R S))),
    P''.IsPrime →
      P''.LiesOver P →
        e₀ ∉ P'' → P'' = Ideal.comap (TensorProduct.map (AlgHom.id R' R') (integralClosure R S).val).toRingHom P'
H : ∀ (P'' : Ideal (R' ⊗[R] S)), P''.IsPrime → P''.LiesOver P → e ∉ P'' → P'' = P'
g : ↥(integralClosure R S)
hgq : ↑g ∉ q
hg : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom g)
f : R'
hfP : f ∉ P
hf :
  Module.Finite (Localization.Away f)
    (Localization.Away ((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)) e))
Pf : Ideal (Localization.Away f) := Ideal.map (algebraMap R' (Localization.Away f)) P
this✝⁸ : Pf.IsPrime
this✝⁷ : 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✝⁶ : Algebra (R' ⊗[R] S) (Localization.Away f ⊗[R] S) := φ.toAlgebra
this✝⁵ : IsScalarTower R' (R' ⊗[R] S) (Localization.Away f ⊗[R] S) :=
  IsScalarTower.of_algebraMap_eq' (Eq.symm (AlgHom.comp_algebraMap φ))
this✝⁴ : IsLocalization.Away (f ⊗ₜ[R] 1) (Localization.Away f ⊗[R] S)
P'f : Ideal (Localization.Away f ⊗[R] S) := Ideal.map (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S)) P'
hP'f : Disjoint ↑(Submonoid.powers (f ⊗ₜ[R] 1)) ↑P'
this✝³ : P'f.IsPrime
this✝² : P'f.LiesOver P'
this✝¹ : P'f.LiesOver P
this✝ : P'f.LiesOver Pf
P'' : Ideal (Localization.Away f ⊗[R] S)
a✝¹ : P''.IsPrime
a✝ : P''.LiesOver Pf
hP'' : (TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)) e ∉ P''
this : P''.LiesOver P
⊢ P'' = P'f

Tactic:
refine congr($(PrimeSpectrum.localization_comap_injective (R := R' ⊗[R] S)
      (Localization.Away f ⊗[R] S) (.powers (f ⊗ₜ 1)) (a₁ := ⟨P'', ‹_›⟩)
      (a₂ := ⟨P'f, ‹_›⟩) (PrimeSpectrum.ext ?_)).1)

Goal state after:
case refine_4
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : FiniteType R S
p : Ideal R
inst✝³ : p.IsPrime
q : Ideal S
inst✝² : q.IsPrime
inst✝¹ : q.LiesOver p
inst✝ : QuasiFiniteAt R q
R' : Type u
w✝⁶ : CommRing R'
w✝⁵ : Algebra R R'
w✝⁴ : Etale R R'
P : Ideal R'
w✝³ : P.IsPrime
w✝² : P.LiesOver p
e : R' ⊗[R] S
he : IsIdempotentElem e
e₀ : R' ⊗[R] ↥(integralClosure R S)
he₀ : IsIdempotentElem e₀
he₀e : (TensorProduct.map (AlgHom.id R' R') (integralClosure R S).val) e₀ = e
P' : Ideal (R' ⊗[R] S)
w✝¹ : P'.IsPrime
w✝ : P'.LiesOver P
hP'q : Ideal.comap TensorProduct.includeRight.toRingHom P' = q
heP' : e ∉ P'
hpP : Function.Bijective ⇑(Ideal.ResidueField.mapₐ p P (ofId R R') ⋯)
H' :
  ∀ (P'' : Ideal (R' ⊗[R] ↥(integralClosure R S))),
    P''.IsPrime →
      P''.LiesOver P →
        e₀ ∉ P'' → P'' = Ideal.comap (TensorProduct.map (AlgHom.id R' R') (integralClosure R S).val).toRingHom P'
H : ∀ (P'' : Ideal (R' ⊗[R] S)), P''.IsPrime → P''.LiesOver P → e ∉ P'' → P'' = P'
g : ↥(integralClosure R S)
hgq : ↑g ∉ q
hg : Function.Bijective ⇑(Localization.awayMap (integralClosure R S).val.toRingHom g)
f : R'
hfP : f ∉ P
hf :
  Module.Finite (Localization.Away f)
    (Localization.Away ((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)) e))
Pf : Ideal (Localization.Away f) := Ideal.map (algebraMap R' (Localization.Away f)) P
this✝⁸ : Pf.IsPrime
this✝⁷ : 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✝⁶ : Algebra (R' ⊗[R] S) (Localization.Away f ⊗[R] S) := φ.toAlgebra
this✝⁵ : IsScalarTower R' (R' ⊗[R] S) (Localization.Away f ⊗[R] S) :=
  IsScalarTower.of_algebraMap_eq' (Eq.symm (AlgHom.comp_algebraMap φ))
this✝⁴ : IsLocalization.Away (f ⊗ₜ[R] 1) (Localization.Away f ⊗[R] S)
P'f : Ideal (Localization.Away f ⊗[R] S) := Ideal.map (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S)) P'
hP'f : Disjoint ↑(Submonoid.powers (f ⊗ₜ[R] 1)) ↑P'
this✝³ : P'f.IsPrime
this✝² : P'f.LiesOver P'
this✝¹ : P'f.LiesOver P
this✝ : P'f.LiesOver Pf
P'' : Ideal (Localization.Away f ⊗[R] S)
a✝¹ : P''.IsPrime
a✝ : P''.LiesOver Pf
hP'' : (TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)) e ∉ P''
this : P''.LiesOver P
⊢ (PrimeSpectrum.comap (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S))
        { asIdeal := P'', isPrime := a✝¹ }).asIdeal =
    (PrimeSpectrum.comap (algebraMap (R' ⊗[R] S) (Localization.Away f ⊗[R] S))
        { asIdeal := P'f, isPrime := this✝³ }).asIdeal