-- 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.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
y : Q.Ring ⊗[P.Ring] ↥P.ker
simp :
  HSMul.hSMul (α := Q.Ring) (β := TensorProduct.{u_2, u_3, u} S T P.Cotangent)
      (↑((e : (a : Q.Ring ⊗[P.Ring] ↥P.ker) → ↥Q.ker) x) : Q.Ring)
      ((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) :
            (a : Q.Ring ⊗[P.Ring] ↥P.ker) → T ⊗[S] P.Cotangent)
          y :
        TensorProduct.{u_2, u_3, u} S T P.Cotangent) =
    0
⊢ 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))

Tactic:
simp only [AddMonoidHom.mem_ker, AddMonoidHom.coe_coe, map_smul,
      LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
      LinearEquiv.symm_apply_apply, f']

Premises used: AddMonoidHom.coe_coe, LinearEquiv.symm_apply_apply, LinearMap.coe_comp, Function.comp_apply, AddMonoidHom.mem_ker, LinearEquiv.coe_coe, map_smul