-- Mathlib/RingTheory/Etale/Kaehler.lean
lemma tensorH1CotangentOfIsLocalization_toLinearMap
    (M : Submonoid S) [IsLocalization M T] :
    (tensorH1CotangentOfIsLocalization R T M).toLinearMap =
      (Algebra.H1Cotangent.map R R S T).liftBaseChange T

Proof state:
case a.h.h
R S T : Type u
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
x : H1Cotangent R S
P : Extension R S := (Generators.self R S).toExtension
M' : Submonoid P.Ring := Submonoid.comap (algebraMap P.Ring S) M
⊢ (Extension.H1Cotangent.equivOfFormallySmooth (Extension.ofSurjective (IsLocalization.liftAlgHom ⋯) ⋯)
        (Generators.self R T).toExtension)
      ((Extension.H1Cotangent.map
          {
            toRingHom :=
              algebraMap (Generators.self R S).Ring
                (Localization (Submonoid.comap (algebraMap (Generators.self R S).Ring S) M)),
            toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ })
        x) =
    (H1Cotangent.map R R S T) x

Tactic:
letI fQ : Localization M' →ₐ[R] T := IsLocalization.liftAlgHom (M := M')
  (f := (IsScalarTower.toAlgHom R S T).comp (IsScalarTower.toAlgHom R P.Ring S)) (fun ⟨y, hy⟩ ↦
  by simpa using IsLocalization.map_units T ⟨algebraMap P.Ring S y, hy⟩)

Result:
case a.h.h
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
⊢ (Extension.H1Cotangent.equivOfFormallySmooth (Extension.ofSurjective (IsLocalization.liftAlgHom ⋯) ⋯)
        (Generators.self R T).toExtension)
      ((Extension.H1Cotangent.map
          {
            toRingHom :=
              algebraMap (Generators.self R S).Ring
                (Localization (Submonoid.comap (algebraMap (Generators.self R S).Ring S) M)),
            toRingHom_algebraMap := ⋯, algebraMap_toRingHom := ⋯ })
        x) =
    (H1Cotangent.map R R S T) x

Proof state:
case intro.intro.mk.intro.intro
R S T : Type u
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
x✝ : H1Cotangent R S
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 ⋯
x s : P.Ring
hs : (algebraMap P.Ring S) s ∈ M
⊢ (IsLocalization.lift ⋯) (IsLocalization.mk' (Localization M') x ⟨s, ⋯⟩) =
    IsLocalization.mk' T ((algebraMap P.Ring S) x) ⟨(algebraMap P.Ring S) s, hs⟩

Tactic:
rw [IsLocalization.lift_mk'_spec]

Result:
case intro.intro.mk.intro.intro
R S T : Type u
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
x✝ : H1Cotangent R S
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 ⋯
x s : P.Ring
hs : (algebraMap P.Ring S) s ∈ M
⊢ ↑((IsScalarTower.toAlgHom R S T).comp (IsScalarTower.toAlgHom R P.Ring S)) x =
    ↑((IsScalarTower.toAlgHom R S T).comp (IsScalarTower.toAlgHom R P.Ring S)) ↑⟨s, ⋯⟩ *
      IsLocalization.mk' T ((algebraMap P.Ring S) x) ⟨(algebraMap P.Ring S) s, hs⟩

Proof state:
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
⊢ (IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ) =
    IsScalarTower.toAlgHom R (Generators.self R T).toExtension.Ring T

Tactic:
ext i

Result:
case hf
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
i : T
⊢ ((IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ)) (MvPolynomial.X i) =
    (IsScalarTower.toAlgHom R (Generators.self R T).toExtension.Ring T) (MvPolynomial.X i)

Proof state:
case hf
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
i : T
⊢ ((IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ)) (MvPolynomial.X i) =
    (algebraMap (Generators.self R T).Ring T) (MvPolynomial.X i)

Tactic:
simp

Result:
no goals

Proof state:
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
⊢ ∀ (x : (Generators.self R T).toExtension.Ring),
    (algebraMap Q.Ring T) ((MvPolynomial.aeval Q.σ).toRingHom x) =
      (algebraMap T T) ((algebraMap (Generators.self R T).toExtension.Ring T) x)

Tactic:
have : (IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ) =
    IsScalarTower.toAlgHom _ (Generators.self R T).toExtension.Ring _ := by
  ext i
  show _ = algebraMap (Generators.self R T).Ring _ (.X i)
  simp

Result:
R S T : Type u
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
x : H1Cotangent R S
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 ⋯
this :
  (IsScalarTower.toAlgHom R Q.Ring T).comp (MvPolynomial.aeval Q.σ) =
    IsScalarTower.toAlgHom R (Generators.self R T).toExtension.Ring T
⊢ ∀ (x : (Generators.self R T).toExtension.Ring),
    (algebraMap Q.Ring T) ((MvPolynomial.aeval Q.σ).toRingHom x) =
      (algebraMap T T) ((algebraMap (Generators.self R T).toExtension.Ring T) x)