-- Module: Mathlib.Algebra.MvPolynomial.Derivation
Goal:
σ : Type u_1
R : Type u_2
A : Type u_3
inst : CommSemiring R
inst_1 : AddCommMonoid A
inst_2 : Module R A
inst_3 : Module (MvPolynomial σ R) A
inst_4 : IsScalarTower R (MvPolynomial σ R) A
D : MvPolynomial σ R →ₗ[R] A
h₁ : Eq.{u_3 + 1} (α := A) ((D : (a : MvPolynomial σ R) → A) 1 : A) (0 : A)
H :
  ∀ (s : σ →₀ ℕ) (i : σ),
    (D : (a : MvPolynomial σ R) → A)
        (HMul.hMul (α := MvPolynomial σ R) ((monomial s : (a : R) → MvPolynomial σ R) 1) (X i)) =
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((monomial s : (a : R) → MvPolynomial σ R) 1)
          ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i)
          ((D : (a : MvPolynomial σ R) → A) ((monomial s : (a : R) → MvPolynomial σ R) 1) : A)
hC :
  ∀ (r : R),
    Eq.{u_3 + 1} (α := A)
      ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) r : MvPolynomial σ R) : A)
      (0 : A)
this :
  ∀ (p : MvPolynomial σ R) (i : σ),
    (D : (a : MvPolynomial σ R) → A) (p * X i) =
      HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i) ((D : (a : MvPolynomial σ R) → A) p : A)
p : MvPolynomial σ R
c : R
C :
  (D : (a : MvPolynomial σ R) → A)
      (HMul.hMul (α := MvPolynomial σ R) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) p) =
    HSMul.hSMul (β := A) p
        ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
        ((D : (a : MvPolynomial σ R) → A) p : A)
C₁ :
  (D : (a : MvPolynomial σ R) → A) (c • p) =
    HSMul.hSMul (β := A) p
        ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
        ((D : (a : MvPolynomial σ R) → A) p : A)
C₂ :
  (D : (a : MvPolynomial σ R) → A) (c • p) =
    HSMul.hSMul (β := A) p 0 +
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
        ((D : (a : MvPolynomial σ R) → A) p : A)
C₃ :
  (D : (a : MvPolynomial σ R) → A) (c • p) =
    0 +
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
        ((D : (a : MvPolynomial σ R) → A) p : A)
C₄ :
  (D : (a : MvPolynomial σ R) → A) (c • p) =
    HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
      ((D : (a : MvPolynomial σ R) → A) p : A)
C₅ :
  HSMul.hSMul (β := A) c ((D : (a : MvPolynomial σ R) → A) p : A) =
    HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
      ((D : (a : MvPolynomial σ R) → A) p : A)
C₆ :
  HSMul.hSMul (β := A) c ((D : (a : MvPolynomial σ R) → A) p : A) =
    HSMul.hSMul (β := A) (HSMul.hSMul (β := MvPolynomial σ R) c 1) ((D : (a : MvPolynomial σ R) → A) p : A)
C₇ C₈ :
  HSMul.hSMul (β := A) c ((D : (a : MvPolynomial σ R) → A) p : A) =
    HSMul.hSMul (β := A) c ((D : (a : MvPolynomial σ R) → A) p : A)
⊢ (D : (a : MvPolynomial σ R) → A)
      (HMul.hMul (β := MvPolynomial σ R) p ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)) =
    HSMul.hSMul (β := A) p
        ((D : (a : MvPolynomial σ R) → A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (β := A) ((MvPolynomial.C : (a : R) → MvPolynomial σ R) c : MvPolynomial σ R)
        ((D : (a : MvPolynomial σ R) → A) p : A)

Tactic:
rw [mul_comm, C_mul', hC, smul_zero, zero_add, D.map_smul, C_eq_smul_one, smul_one_smul]

Premises used: smul_zero, zero_add, MvPolynomial.C_mul', smul_one_smul, LinearMap.map_smul, mul_comm, MvPolynomial.C_eq_smul_one