Module: Mathlib.RingTheory.QuasiFinite.Basic

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R q
e : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}) ≃ₜ
  PrimeSpectrum ({ asIdeal := p, isPrime := ⋯ }.asIdeal.Fiber S) :=
  PrimeSpectrum.preimageHomeomorphFiber R S { asIdeal := p, isPrime := ⋯ }
qF : PrimeSpectrum (p.Fiber S) := e ⟨{ asIdeal := q, isPrime := inst✝³ }, ⋯⟩
this✝ : QuasiFiniteAt p.ResidueField qF.asIdeal
r : p.Fiber S
hr : r ∉ qF.asIdeal
hrq : ↑(PrimeSpectrum.basicOpen r) = {{ asIdeal := qF.asIdeal, isPrime := ⋯ }}
s : R
hs : s ∉ p
x : S
hsx : s • r = 1 ⊗ₜ[R] x
this : x ∉ q
q' : Ideal S
x✝¹ : q'.IsPrime
hq' : q' ≠ q
x✝ : q'.LiesOver p
hxq' : x ∉ q'
⊢ r ∉
    RingHom.ker
      (TensorProduct.lift
          (Ideal.ResidueField.mapₐ { asIdeal := p, isPrime := ⋯ }.asIdeal
            (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal (ofId R S) ⋯)
          (IsScalarTower.toAlgHom R S (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal.ResidueField) ⋯).toRingHom

Tactic:
← Ideal.IsPrime.mul_mem_left_iff (x := algebraMap _ _ s),

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R q
e : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}) ≃ₜ
  PrimeSpectrum ({ asIdeal := p, isPrime := ⋯ }.asIdeal.Fiber S) :=
  PrimeSpectrum.preimageHomeomorphFiber R S { asIdeal := p, isPrime := ⋯ }
qF : PrimeSpectrum (p.Fiber S) := e ⟨{ asIdeal := q, isPrime := inst✝³ }, ⋯⟩
this✝ : QuasiFiniteAt p.ResidueField qF.asIdeal
r : p.Fiber S
hr : r ∉ qF.asIdeal
hrq : ↑(PrimeSpectrum.basicOpen r) = {{ asIdeal := qF.asIdeal, isPrime := ⋯ }}
s : R
hs : s ∉ p
x : S
hsx : s • r = 1 ⊗ₜ[R] x
this : x ∉ q
q' : Ideal S
x✝¹ : q'.IsPrime
hq' : q' ≠ q
x✝ : q'.LiesOver p
hxq' : x ∉ q'
⊢ (algebraMap R (p.Fiber S)) s * r ∉
    RingHom.ker
      (TensorProduct.lift
          (Ideal.ResidueField.mapₐ { asIdeal := p, isPrime := ⋯ }.asIdeal
            (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal (ofId R S) ⋯)
          (IsScalarTower.toAlgHom R S (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal.ResidueField) ⋯).toRingHom

R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
inst✝⁴ : p.IsPrime
q : Ideal S
inst✝³ : q.IsPrime
inst✝² : q.LiesOver p
inst✝¹ : EssFiniteType R S
inst✝ : QuasiFiniteAt R q
e : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isPrime := ⋯ }}) ≃ₜ
  PrimeSpectrum ({ asIdeal := p, isPrime := ⋯ }.asIdeal.Fiber S) :=
  PrimeSpectrum.preimageHomeomorphFiber R S { asIdeal := p, isPrime := ⋯ }
qF : PrimeSpectrum (p.Fiber S) := e ⟨{ asIdeal := q, isPrime := inst✝³ }, ⋯⟩
this✝ : QuasiFiniteAt p.ResidueField qF.asIdeal
r : p.Fiber S
hr : r ∉ qF.asIdeal
hrq : ↑(PrimeSpectrum.basicOpen r) = {{ asIdeal := qF.asIdeal, isPrime := ⋯ }}
s : R
hs : s ∉ p
x : S
hsx : s • r = 1 ⊗ₜ[R] x
this : x ∉ q
q' : Ideal S
x✝¹ : q'.IsPrime
hq' : q' ≠ q
x✝ : q'.LiesOver p
hxq' : x ∉ q'
⊢ (algebraMap R (p.Fiber S)) s ∉
    RingHom.ker
      (TensorProduct.lift
          (Ideal.ResidueField.mapₐ { asIdeal := p, isPrime := ⋯ }.asIdeal
            (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal (ofId R S) ⋯)
          (IsScalarTower.toAlgHom R S (↑⟨{ asIdeal := q', isPrime := x✝¹ }, ⋯⟩).asIdeal.ResidueField) ⋯).toRingHom