Module: Mathlib.RingTheory.Etale.QuasiFinite

Goal state before:
pp: 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 : Pf.IsPrime
⊢ ∃ R' x x_1,
    ∃ (_ : Etale R R'),
      ∃ P,
        ∃ (x_3 : P.IsPrime) (x_4 : P.LiesOver p),
          ∃ e,
            ∃ (_ : IsIdempotentElem e),
              ∃ P',
                ∃ (_ : P'.IsPrime) (_ : P'.LiesOver P),
                  Ideal.comap TensorProduct.includeRight.toRingHom P' = q ∧
                    e ∉ P' ∧
                      Function.Bijective ⇑(Ideal.ResidueField.mapₐ p P (ofId R R') ⋯) ∧
                        Module.Finite R' (Localization.Away e) ∧
                          ∀ (P'' : Ideal (R' ⊗[R] S)), P''.IsPrime → P''.LiesOver P → e ∉ P'' → P'' = P'

Tactic:
have : Pf.LiesOver P := IsLocalization.liesOver_map_of_isPrime_disjoint (.powers f) _ (by
    rwa [Ideal.disjoint_powers_iff_notMem _ (Ideal.IsPrime.isRadical ‹_›)])

Goal state after:
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✝ : Pf.IsPrime
this : Pf.LiesOver P
⊢ ∃ R' x x_1,
    ∃ (_ : Etale R R'),
      ∃ P,
        ∃ (x_3 : P.IsPrime) (x_4 : P.LiesOver p),
          ∃ e,
            ∃ (_ : IsIdempotentElem e),
              ∃ P',
                ∃ (_ : P'.IsPrime) (_ : P'.LiesOver P),
                  Ideal.comap TensorProduct.includeRight.toRingHom P' = q ∧
                    e ∉ P' ∧
                      Function.Bijective ⇑(Ideal.ResidueField.mapₐ p P (ofId R R') ⋯) ∧
                        Module.Finite R' (Localization.Away e) ∧
                          ∀ (P'' : Ideal (R' ⊗[R] S)), P''.IsPrime → P''.LiesOver P → e ∉ P'' → P'' = P'