-- Module: Mathlib.RingTheory.Etale.QuasiFinite
Goal:
R : Type u_2
S : Type u_3
R' : Type u_4
R'' : Type u_5
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : FiniteType R S
inst_4 : CommRing R'
inst_5 : Algebra R R'
inst_6 : CommRing R''
inst_7 : Algebra R R''
inst_8 : Algebra R'' S
inst_9 : Algebra.IsIntegral R R''
inst_10 : IsScalarTower R R'' S
q : Ideal S
P : Ideal R'
inst_11 : P.IsPrime
e : R' ⊗[R] S
e₀ : R' ⊗[R] R''
he₀ : IsIdempotentElem e₀
he₀e :
  ((TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
        (a : R' ⊗[R] R'') → R' ⊗[R] S)
      e₀ =
    e
P' : Ideal (R' ⊗[R] S)
hP'q : Ideal.comap (F := S →+* R' ⊗[R] S) (AlgHom.toRingHom TensorProduct.includeRight) P' = q
H :
  ∀ (P'' : Ideal (R' ⊗[R] R'')),
    P''.IsPrime →
      P''.LiesOver P →
        e₀ ∉ P'' →
          P'' =
            Ideal.comap (F := R' ⊗[R] R'' →+* R' ⊗[R] S)
              (AlgHom.toRingHom
                (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S))
              P'
g : R''
hgq : ((algebraMap R'' S : (a : R'') → S) g : S) ∉ q
hg :
  Function.Surjective.{u_5 + 1, u_3 + 1} (α := Localization.Away g) (β :=
    Localization.Away (M := S) ((algebraMap R'' S : (a : R'') → S) g : S))
    (⇑(Localization.awayMap (algebraMap R'' S) g) :
      (a : Localization.Away g) → Localization.Away ((algebraMap R'' S : (a : R'') → S) g))
this_1 : Algebra.IsIntegral R' (Localization.Away e₀)
f : R'
hfP : f ∉ P
hf :
  AlgHom.Finite (R := R') (A := Localization.Away f) (B :=
    Localization.Away (M := Localization.Away e)
      ((ofId R' (Localization.Away e) : (a : R') → Localization.Away e) f : Localization.Away e))
    (Localization.awayMapₐ (ofId R' (Localization.Away e)) f)
φ : R' ⊗[R] S →ₐ[R'] TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S :=
  TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S)
algInst : Algebra (R' ⊗[R] S) (TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S) := φ.toAlgebra
this_2 : IsLocalization.Away (R := R' ⊗[R] S) (f ⊗ₜ[R] 1) (TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
this_3 :
  IsLocalization.Away (e * f ⊗ₜ[R] 1)
    (Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
      ((φ : (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S) e : TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S))
this :
  IsLocalization.Away (e * f ⊗ₜ[R] 1)
    (Localization.Away (M := Localization.Away e)
      ((ofId R' (Localization.Away e) : (a : R') → Localization.Away e) f : Localization.Away e))
equiv : Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
    ((φ : (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S) e :
      TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S) ≃ₐ[R' ⊗[R] S]
  Localization.Away (M := Localization.Away e)
    ((ofId R' (Localization.Away e) : (a : R') → Localization.Away e) f : Localization.Away e) :=
  IsLocalization.algEquiv (Submonoid.powers (e * f ⊗ₜ[R] 1))
    (Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
      ((φ : (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S) e : TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S))
    (Localization.Away (M := Localization.Away e)
      ((ofId R' (Localization.Away e) : (a : R') → Localization.Away e) f : Localization.Away e))
refine :
  RingHom.Finite (A := Localization.Away f) (B :=
    Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
      (((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S) :
              R' ⊗[R] S →ₐ[R'] TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S) :
            (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S)
          e :
        TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S))
    (algebraMap (Localization.Away f)
      (Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
        (((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S) :
                R' ⊗[R] S →ₐ[R'] TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S) :
              (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S)
            e :
          TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)))
⊢ Module.Finite (Localization.Away f)
    (Localization.Away (M := TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S)
      (((TensorProduct.map (ofId R' (Localization.Away f)) (AlgHom.id R S) :
              R' ⊗[R] S →ₐ[R'] TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S) :
            (a : R' ⊗[R] S) → Localization.Away f ⊗[R] S)
          e :
        TensorProduct.{u_2, u_4, u_3} R (Localization.Away f) S))

Tactic:
refine RingHom.finite_algebraMap.mp ?_

Premises used: RingHom.finite_algebraMap, Iff.mp