-- Module: Mathlib.RingTheory.Kaehler.TensorProduct
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
B : Type u_4
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : CommRing A
inst_4 : CommRing B
inst_5 : Algebra R A
inst_6 : Algebra R B
inst_7 : Algebra A B
inst_8 : Algebra S B
inst_9 : IsScalarTower R A B
inst_10 : IsScalarTower R S B
inst_11 : Algebra.IsPushout R S A B
x : S
x_1 y : A
H :
  x •
      HSMul.hSMul (β := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]) x_1
        (1 ⊗ₜ[R] (D R A : (a : A) → Ω[A⁄R]) y : TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]) =
    x ⊗ₜ[R] HSMul.hSMul (β := Ω[A⁄R]) x_1 ((D R A : (a : A) → Ω[A⁄R]) y : Ω[A⁄R])
⊢ Eq (α := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
    ((((↑R :
                {S : Type u_2} →
                  {M M₂ : Type (max u_2 u_3)} →
                    [inst : Semiring R] →
                      [inst_12 : Semiring S] →
                        [inst_13 : AddCommMonoid M] →
                          [inst_14 : AddCommMonoid M₂] →
                            [inst_15 : Module R M] →
                              [inst_16 : Module R M₂] →
                                [inst_17 : Module S M] →
                                  [inst_18 : Module S M₂] →
                                    [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
              ((↑S :
                    {S_1 M : Type u_4} →
                      {M₂ : Type (max u_2 u_3)} →
                        [inst : Semiring S] →
                          [inst_12 : Semiring S_1] →
                            [inst_13 : AddCommMonoid M] →
                              [inst_14 : AddCommMonoid M₂] →
                                [inst_15 : Module S M] →
                                  [inst_16 : Module S M₂] →
                                    [inst_17 : Module S_1 M] →
                                      [inst_18 : Module S_1 M₂] →
                                        [LinearMap.CompatibleSMul M M₂ S S_1] → (fₗ : M →ₗ[S_1] M₂) → M →ₗ[S] M₂)
                  (Derivation.liftKaehlerDifferential (R := S) (S := B) (M := TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
                    (derivationTensorProduct R S A B)) ∘ₗ
                LinearMap.liftBaseChange S
                  ((↑R :
                      {S M : Type u_3} →
                        {M₂ : Type u_4} →
                          [inst : Semiring R] →
                            [inst_12 : Semiring S] →
                              [inst_13 : AddCommMonoid M] →
                                [inst_14 : AddCommMonoid M₂] →
                                  [inst_15 : Module R M] →
                                    [inst_16 : Module R M₂] →
                                      [inst_17 : Module S M] →
                                        [inst_18 : Module S M₂] →
                                          [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
                    (map R S A B))) :
            TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R] →ₗ[R] TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R]) :
          (a : S ⊗[R] Ω[A⁄R]) → S ⊗[R] Ω[A⁄R])
        (x ⊗ₜ[R] (Derivation.tensorProductTo (D R A) : (a : A ⊗[R] A) → Ω[A⁄R]) (x_1 ⊗ₜ[R] y)) :
      TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])
    (((↑R :
              {S : Type u_2} →
                {M M₂ : Type (max u_2 u_3)} →
                  [inst : Semiring R] →
                    [inst_12 : Semiring S] →
                      [inst_13 : AddCommMonoid M] →
                        [inst_14 : AddCommMonoid M₂] →
                          [inst_15 : Module R M] →
                            [inst_16 : Module R M₂] →
                              [inst_17 : Module S M] →
                                [inst_18 : Module S M₂] →
                                  [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
            (S := S) LinearMap.id :
          (a : S ⊗[R] Ω[A⁄R]) → S ⊗[R] Ω[A⁄R])
        (x ⊗ₜ[R] (Derivation.tensorProductTo (D R A) : (a : A ⊗[R] A) → Ω[A⁄R]) (x_1 ⊗ₜ[R] y)) :
      TensorProduct.{u_1, u_2, u_3} R S Ω[A⁄R])

Tactic:
simp only [LinearMap.restrictScalars_comp, Derivation.tensorProductTo_tmul, LinearMap.coe_comp,
      LinearMap.coe_restrictScalars, Function.comp_apply, LinearMap.liftBaseChange_tmul, map_smul,
      map_D, LinearMap.map_smul_of_tower, Derivation.liftKaehlerDifferential_comp_D,
      LinearMap.id_coe, id_eq, derivationTensorProduct_algebraMap]

Premises used: LinearMap.id_coe, LinearMap.coe_restrictScalars, Derivation.liftKaehlerDifferential_comp_D, LinearMap.restrictScalars_comp, LinearMap.liftBaseChange_tmul, LinearMap.coe_comp, KaehlerDifferential.derivationTensorProduct_algebraMap, Derivation.tensorProductTo_tmul, Function.comp_apply, LinearMap.map_smul_of_tower, KaehlerDifferential.map_D, map_smul, id_eq