Module: Mathlib.RingTheory.Etale.Kaehler

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
M : Submonoid S
inst✝ : IsLocalization M T
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom ⋯
Q : Extension R T := Extension.ofSurjective fQ ⋯
f : P.Hom Q :=
  { toRingHom := algebraMap P.Ring (Localization M'), toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ }
this✝⁸ : FormallySmooth R P.Ring
this✝⁷ : FormallySmooth P.Ring (Localization M')
this✝⁶ : FormallySmooth R Q.Ring
this✝⁵ : Module.Flat S T
this✝⁴ : Algebra P.Ring Q.Ring := inferInstance
this✝³ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝² : Algebra (Localization M') T := fQ.toAlgebra
this✝¹ : IsScalarTower P.Ring S T
this✝ : IsScalarTower P.Ring (Localization M') T
this : IsLocalizedModule M' (IsScalarTower.toAlgHom P.Ring S T).toLinearMap
⊢ f.toRingHom.FormallyEtale

pp: case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
M : Submonoid S
inst✝ : IsLocalization M T
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom ⋯
Q : Extension R T := Extension.ofSurjective fQ ⋯
f : P.Hom Q :=
  { toRingHom := algebraMap P.Ring (Localization M'), toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ }
this✝⁸ : FormallySmooth R P.Ring
this✝⁷ : FormallySmooth P.Ring (Localization M')
this✝⁶ : FormallySmooth R Q.Ring
this✝⁵ : Module.Flat S T
this✝⁴ : Algebra P.Ring Q.Ring := inferInstance
this✝³ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝² : Algebra (Localization M') T := fQ.toAlgebra
this✝¹ : IsScalarTower P.Ring S T
this✝ : IsScalarTower P.Ring (Localization M') T
this : IsLocalizedModule M' (IsScalarTower.toAlgHom P.Ring S T).toLinearMap
⊢ Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer ⋯))

Tactic:
· exact RingHom.formallyEtale_algebraMap.mpr
      (FormallyEtale.of_isLocalization (M := M') (Rₘ := Localization M'))

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R S
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
M : Submonoid S
inst✝ : IsLocalization M T
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom ⋯
Q : Extension R T := Extension.ofSurjective fQ ⋯
f : P.Hom Q :=
  { toRingHom := algebraMap P.Ring (Localization M'), toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ }
this✝⁸ : FormallySmooth R P.Ring
this✝⁷ : FormallySmooth P.Ring (Localization M')
this✝⁶ : FormallySmooth R Q.Ring
this✝⁵ : Module.Flat S T
this✝⁴ : Algebra P.Ring Q.Ring := inferInstance
this✝³ : Algebra P.Ring T := ((algebraMap S T).comp (algebraMap P.Ring S)).toAlgebra
this✝² : Algebra (Localization M') T := fQ.toAlgebra
this✝¹ : IsScalarTower P.Ring S T
this✝ : IsScalarTower P.Ring (Localization M') T
this : IsLocalizedModule M' (IsScalarTower.toAlgHom P.Ring S T).toLinearMap
⊢ Function.Bijective ⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer ⋯))