-- Module: Mathlib.RingTheory.Kaehler.JacobiZariski
Goal:
R : Type u₁
S : Type u₂
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
T : Type u₃
inst_3 : CommRing T
inst_4 : Algebra R T
inst_5 : Algebra S T
inst_6 : IsScalarTower R S T
ι : Type w₁
σ : Type w₂
Q : Generators S T ι
P : Generators R S σ
this_1 : AddCommGroup (T ⊗[S] Ω[S⁄R]) := inferInstance
this : IsScalarTower (Generators.Ring (Q.comp P)) (Generators.Ring (Q.comp P)) T
p : MvPolynomial (ι ⊕ σ) R
IH :
  Eq (α := T ⊗[S] Ω[S⁄R])
    ((δAux R Q : (a : Q.Ring) → T ⊗[S] Ω[S⁄R]) ((Hom.toAlgHom (Q.ofComp P) : (a : (Q.comp P).Ring) → Q.Ring) p) :
      T ⊗[S] Ω[S⁄R])
    ((LinearMap.baseChange T P.toExtension.toKaehler : (a : T ⊗[S] P.toExtension.CotangentSpace) → T ⊗[S] Ω[S⁄R])
        ((CotangentSpace.compEquiv Q P :
              (a : (Q.comp P).toExtension.CotangentSpace) →
                Q.toExtension.CotangentSpace × T ⊗[S] P.toExtension.CotangentSpace)
            (1 ⊗ₜ[Generators.Ring (Q.comp P)]
              (D R (Generators.Ring (Q.comp P)) : (a : (Q.comp P).Ring) → Ω[(Q.comp P).Ring⁄R]) p)).2 :
      T ⊗[S] Ω[S⁄R])
n : σ
mul_X :
  HSMul.hSMul (β := T ⊗[S] Ω[S⁄R])
      (eval₂ (algebraMap R T) (fun (i : ι ⊕ σ) ↦ Sum.elim Q.val ((⇑(algebraMap S T) : (a : S) → T) ∘ P.val) i) p)
      (1 ⊗ₜ[S] (D R S : (a : S) → Ω[S⁄R]) (P.val n) : T ⊗[S] Ω[S⁄R]) =
    HSMul.hSMul (α := T) (β := T ⊗[S] Ω[S⁄R]) ((aeval (val (Q.comp P)) : (a : MvPolynomial (ι ⊕ σ) R) → T) p : T)
      (1 ⊗ₜ[S] (D R S : (a : S) → Ω[S⁄R]) (P.val n) : T ⊗[S] Ω[S⁄R])
⊢ HSMul.hSMul (β := T ⊗[S] Ω[S⁄R])
        (eval₂ (algebraMap R T) (fun (i : ι ⊕ σ) ↦ Sum.elim Q.val ((⇑(algebraMap S T) : (a : S) → T) ∘ P.val) i) p)
        ((δAux R Q : (a : Q.Ring) → T ⊗[S] Ω[S⁄R])
            (Sum.elim X ((⇑C : (a : S) → MvPolynomial ι S) ∘ P.val) (Sum.inr n)) :
          T ⊗[S] Ω[S⁄R]) +
      HSMul.hSMul (α := T) (β := T ⊗[S] Ω[S⁄R])
        ((aeval Q.val : (a : MvPolynomial ι S) → T)
            (Sum.elim X ((⇑C : (a : S) → MvPolynomial ι S) ∘ P.val) (Sum.inr n)) :
          T)
        ((LinearMap.baseChange T P.toExtension.toKaehler : (a : T ⊗[S] P.toExtension.CotangentSpace) → T ⊗[S] Ω[S⁄R])
          ((CotangentSpace.compEquiv Q P :
                (a : (Q.comp P).toExtension.CotangentSpace) →
                  Q.toExtension.CotangentSpace × T ⊗[S] P.toExtension.CotangentSpace)
              (1 ⊗ₜ[Generators.Ring (Q.comp P)]
                (D R (Generators.Ring (Q.comp P)) : (a : (Q.comp P).Ring) → Ω[(Q.comp P).Ring⁄R]) p)).2) =
    HSMul.hSMul (α := T) (β := T ⊗[S] Ω[S⁄R]) ((aeval (val (Q.comp P)) : (a : MvPolynomial (ι ⊕ σ) R) → T) p : T)
        ((LinearMap.baseChange T P.toExtension.toKaehler : (a : T ⊗[S] P.toExtension.CotangentSpace) → T ⊗[S] Ω[S⁄R])
          ((CotangentSpace.compEquiv Q P :
                (a : (Q.comp P).toExtension.CotangentSpace) →
                  Q.toExtension.CotangentSpace × T ⊗[S] P.toExtension.CotangentSpace)
              ((cotangentSpaceBasis (Q.comp P) : (a : ι ⊕ σ) → (Q.comp P).toExtension.CotangentSpace) (Sum.inr n))).2) +
      HSMul.hSMul (β := T ⊗[S] Ω[S⁄R]) (Sum.elim Q.val ((⇑(algebraMap S T) : (a : S) → T) ∘ P.val) (Sum.inr n))
        ((LinearMap.baseChange T P.toExtension.toKaehler : (a : T ⊗[S] P.toExtension.CotangentSpace) → T ⊗[S] Ω[S⁄R])
          ((CotangentSpace.compEquiv Q P :
                (a : (Q.comp P).toExtension.CotangentSpace) →
                  Q.toExtension.CotangentSpace × T ⊗[S] P.toExtension.CotangentSpace)
              (1 ⊗ₜ[Generators.Ring (Q.comp P)]
                (D R (Generators.Ring (Q.comp P)) : (a : (Q.comp P).Ring) → Ω[(Q.comp P).Ring⁄R]) p)).2)

Tactic:
simp only [Sum.elim_inr, Function.comp_apply, algHom_C, δAux_C,
        CotangentSpace.compEquiv, LinearEquiv.trans_apply, Basis.repr_symm_apply,
        algebraMap_smul, Basis.repr_self, Finsupp.linearCombination_single, Basis.prod_apply,
        LinearMap.coe_inr, Basis.baseChange_apply, one_smul, LinearMap.baseChange_tmul,
        toKaehler_cotangentSpaceBasis, add_left_inj, LinearMap.coe_inl]

Premises used: Algebra.Generators.CotangentSpace.compEquiv, LinearMap.coe_inr, Algebra.Generators.H1Cotangent.δAux_C, Module.Basis.prod_apply, LinearEquiv.trans_apply, algebraMap_smul, add_left_inj, Module.Basis.baseChange_apply, MvPolynomial.algHom_C, Finsupp.linearCombination_single, one_smul, Sum.elim_inr, LinearMap.coe_inl, Algebra.Generators.toKaehler_cotangentSpaceBasis, Function.comp_apply, Module.Basis.repr_symm_apply, LinearMap.baseChange_tmul, Module.Basis.repr_self