awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
H : f.toRingHom.FormallyEtale
this_1 : Algebra P.Ring Q.Ring := f.toRingHom.toAlgebra
this_2 : IsScalarTower.{u_1, u, u} R P.Ring Q.Ring
this_3 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_4 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this_5 : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
this : FormallyEtale.{u, u} P.Ring Q.Ring
x_1 : Q.CotangentSpace
a_1 : Q.Ring
x : Q.Ring ⊗[P.Ring] Ω[P.Ring⁄R]
hx :
  Eq (α := Ω[Q.Ring⁄R])
    ((tensorKaehlerEquivOfFormallyEtale R P.Ring Q.Ring : (a : Q.Ring ⊗[P.Ring] Ω[P.Ring⁄R]) → Ω[Q.Ring⁄R]) x :
      Ω[Q.Ring⁄R])
    ((D R Q.Ring : (a : Q.Ring) → Ω[Q.Ring⁄R]) a_1 : Ω[Q.Ring⁄R])
a : P.Ring
⊢ Eq (α := Q.CotangentSpace)
    (1 ⊗ₜ[Q.Ring] (D R Q.Ring : (a : Q.Ring) → Ω[Q.Ring⁄R]) ((f.toRingHom : (a : P.Ring) → Q.Ring) a) :
      TensorProduct.{u, u_3, u} Q.Ring T Ω[Q.Ring⁄R])
    (1 ⊗ₜ[Q.Ring] (D R Q.Ring : (a : Q.Ring) → Ω[Q.Ring⁄R]) ((algebraMap P.Ring Q.Ring : (a : P.Ring) → Q.Ring) a) :
      TensorProduct.{u, u_3, u} Q.Ring T Ω[Q.Ring⁄R])

Tactic:
rfl
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
x : ↥Q.ker
hx : x ∈ (HSMul.hSMul (β := Submodule Q.Ring ↥Q.ker) Q.ker ⊤).toAddSubgroup
⊢ Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) x

Tactic:
Submodule.smul_induction_on hx ?_ fun _ _ ↦ add_mem

Premises used: Submodule.smul_induction_on
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
x : ↥Q.ker
hx : x ∈ (HSMul.hSMul (β := Submodule Q.Ring ↥Q.ker) Q.ker ⊤).toAddSubgroup
refine :
  ∀ r ∈ Q.ker,
    ∀ (n : ↥Q.ker),
      Membership.mem (γ := Submodule Q.Ring ↥Q.ker) ⊤ n →
        Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) (r • n)
⊢ Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) x

Tactic:
refine Submodule.smul_induction_on hx ?_ fun _ _ ↦ add_mem
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
a : Q.Ring
ha : a ∈ Q.ker
b : ↥Q.ker
obtain :
  ∀ (x : Q.Ring ⊗[P.Ring] ↥P.ker)
    (hx : (e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x = Subtype.mk (p := fun (x : Q.Ring) ↦ x ∈ Q.ker) a ha),
    Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) (a • b)
⊢ Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) (a • b)

Tactic:
obtain ⟨x, hx⟩ := e.surjective ⟨a, ha⟩

Premises used: LinearEquiv.surjective, Subtype.mk
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
a : Q.Ring
ha : a ∈ Q.ker
b : ↥Q.ker
x : Q.Ring ⊗[P.Ring] ↥P.ker
hx : (e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x = Subtype.mk (p := fun (x : Q.Ring) ↦ x ∈ Q.ker) a ha
⊢ (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) = a

Tactic:
congr_arg Subtype.val hx

Premises used: congr_arg, Subtype.val
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
a : Q.Ring
ha : a ∈ Q.ker
b : ↥Q.ker
x : Q.Ring ⊗[P.Ring] ↥P.ker
hx : (e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x = Subtype.mk (p := fun (x : Q.Ring) ↦ x ∈ Q.ker) a ha
obtain :
  ∀ (ha : (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) ∈ Q.ker)
    (hx :
      (e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x =
        Subtype.mk (p := fun (x : Q.Ring) ↦ x ∈ Q.ker) (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) ha),
    Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent))
      (HSMul.hSMul (α := Q.Ring) (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) b)
⊢ Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent)) (a • b)

Tactic:
obtain rfl : (e x).1 = a := congr_arg Subtype.val hx

Premises used: congr_arg, Subtype.val, DFunLike.coe, rfl
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
b : ↥Q.ker
x : Q.Ring ⊗[P.Ring] ↥P.ker
ha : (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) ∈ Q.ker
hx :
  (e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x =
    Subtype.mk (p := fun (x : Q.Ring) ↦ x ∈ Q.ker) (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) ha
obtain :
  ∀ (y : Q.Ring ⊗[P.Ring] ↥P.ker),
    Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent))
      (HSMul.hSMul (α := Q.Ring) (β := ↥Q.ker) (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring)
        ((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) y : ↥Q.ker))
⊢ Membership.mem (γ := AddSubgroup ↥Q.ker) (AddMonoidHom.ker (↑f' : ↥Q.ker →+ T ⊗[S] P.Cotangent))
    (HSMul.hSMul (α := Q.Ring) (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring) b)

Tactic:
obtain ⟨y, rfl⟩ := e.surjective b

Premises used: LinearEquiv.surjective, rfl
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
a : Q.Ring
b : ↥P.ker
c : Q.Ring
d : ↥P.ker
⊢ (HSMul.hSMul (β := P.Cotangent) (↑b : P.Ring) ((Cotangent.mk : (a : ↥P.ker) → P.Cotangent) d : P.Cotangent)).val =
    Cotangent.val 0

Tactic:
simp [Cotangent.smul_eq_zero_of_mem _ b.2]

Premises used: Algebra.Extension.Cotangent.smul_eq_zero_of_mem, Subtype.property
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this_2 : IsScalarTower.{u, u_2, u_3} P.Ring S T
this : IsScalarTower.{u, u, u_3} P.Ring Q.Ring T
e : Q.Ring ⊗[P.Ring] ↥P.ker ≃ₗ[Q.Ring] ↥Q.ker :=
  LinearEquiv.ofBijective (LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) H
f' : LinearMap (R := Q.Ring) (RingHom.id Q.Ring) (↥Q.ker) (TensorProduct.{u_2, u_3, u} S T P.Cotangent) :=
  LinearMap.liftBaseChange Q.Ring
      ((↑P.Ring :
            {S_1 : Type u_2} →
              {M : Type u} →
                {M₂ : Type (max u_3 u)} →
                  [inst_7 : Semiring P.Ring] →
                    [inst_8 : Semiring S_1] →
                      [inst_9 : AddCommMonoid M] →
                        [inst_10 : AddCommMonoid M₂] →
                          [inst_11 : Module P.Ring M] →
                            [inst_12 : Module P.Ring M₂] →
                              [inst_13 : Module S_1 M] →
                                [inst_14 : Module S_1 M₂] →
                                  [LinearMap.CompatibleSMul M M₂ P.Ring S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[P.Ring] M₂)
          (S := S) ((TensorProduct.mk S T P.Cotangent : (a : T) → P.Cotangent →ₗ[S] T ⊗[S] P.Cotangent) 1) ∘ₗ
        Cotangent.mk) ∘ₗ
    (↑e.symm : ↥Q.ker →ₗ[Q.Ring] Q.Ring ⊗[P.Ring] ↥P.ker)
a : Q.Ring
b : ↥P.ker
c : Q.Ring
d : ↥P.ker
⊢ HSMul.hSMul (β := P.Cotangent) (↑b : P.Ring) ((Cotangent.mk : (a : ↥P.ker) → P.Cotangent) d : P.Cotangent) = 0

Tactic:
have h₂ : b.1 • Cotangent.mk d = 0 := by ext; simp [Cotangent.smul_eq_zero_of_mem _ b.2]
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
x : Q.Ring
y : ↥P.ker
this : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
⊢ IsScalarTower.{u, u_2, u_3} P.Ring S T

Tactic:
haveI : IsScalarTower P.Ring S T := .of_algebraMap_eq' rfl
-- Module: Mathlib.RingTheory.Etale.Kaehler
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
inst_5 : Algebra S T
P : Extension.{u, u_1, u_2} R S
Q : Extension.{u, u_1, u_3} R T
f : P.Hom Q
alg : Algebra P.Ring Q.Ring
halg : algebraMap P.Ring Q.Ring = f.toRingHom
H :
  Function.Bijective (α := Q.Ring ⊗[P.Ring] ↥P.ker) (β := ↥Q.ker)
    (⇑(LinearMap.liftBaseChange Q.Ring (f.mapKer halg)) : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker)
x : Q.Ring
y : ↥P.ker
this_1 : Algebra.{u, u_3} P.Ring T :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S T) (algebraMap P.Ring S))
this : IsScalarTower.{u, u_2, u_3} P.Ring S T
⊢ IsScalarTower.{u, u, u_3} P.Ring Q.Ring T

Tactic:
haveI : IsScalarTower P.Ring Q.Ring T :=
    .of_algebraMap_eq fun r ↦ halg ▸ (f.algebraMap_toRingHom r).symm