-- 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) ((C : (a : R) → MvPolynomial σ R) r : MvPolynomial σ R) : A)
      (0 : A)
i : σ
p q : MvPolynomial σ R
hp :
  (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)
hq :
  (D : (a : MvPolynomial σ R) → A) (q * X i) =
    HSMul.hSMul (β := A) q ((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) q : A)
add :
  (D : (a : MvPolynomial σ R) → A) (p * X i + q * X i) =
    HSMul.hSMul (β := A) (p + q) ((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 + q) : A)
add₁ :
  HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) (p * X i) : A)
      ((D : (a : MvPolynomial σ R) → A) (q * X i) : A) =
    HSMul.hSMul (β := A) (p + q) ((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 + q) : A)
add₂ :
  HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) (p * X i) : A)
      ((D : (a : MvPolynomial σ R) → A) (q * X i) : A) =
    HSMul.hSMul (β := A) (p + q) ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (X i)
        (HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) p : A) ((D : (a : MvPolynomial σ R) → A) q : A))
add₃ :
  HAdd.hAdd (β := A)
      (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))
      ((D : (a : MvPolynomial σ R) → A) (q * X i) : A) =
    HSMul.hSMul (β := A) (p + q) ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (X i)
        (HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) p : A) ((D : (a : MvPolynomial σ R) → A) q : A))
add₄ :
  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) +
      (HSMul.hSMul (β := A) q ((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) q : A)) =
    HSMul.hSMul (β := A) (p + q) ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (X i)
        (HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) p : A) ((D : (a : MvPolynomial σ R) → A) q : A))
add₅ :
  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) +
      (HSMul.hSMul (β := A) q ((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) q : A)) =
    HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (β := A) q ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
      HSMul.hSMul (α := MvPolynomial σ R) (X i)
        (HAdd.hAdd (α := A) (β := A) ((D : (a : MvPolynomial σ R) → A) p : A) ((D : (a : MvPolynomial σ R) → A) q : A))
add₆ :
  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) +
      (HSMul.hSMul (β := A) q ((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) q : A)) =
    HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (β := A) q ((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) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i) ((D : (a : MvPolynomial σ R) → A) q : A))
add₇ add₈ :
  HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (β := A) q ((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) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i) ((D : (a : MvPolynomial σ R) → A) q : A)) =
    HSMul.hSMul (β := A) p ((D : (a : MvPolynomial σ R) → A) (X i) : A) +
        HSMul.hSMul (β := A) q ((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) +
        HSMul.hSMul (α := MvPolynomial σ R) (β := A) (X i) ((D : (a : MvPolynomial σ R) → A) q : A))
⊢ (D : (a : MvPolynomial σ R) → A) ((p + q) * X i) =
    HSMul.hSMul (β := A) (p + q) ((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 + q) : A)

Tactic:
rw [add_mul, map_add, map_add, hp, hq, add_smul, smul_add, add_add_add_comm]

Premises used: add_mul, map_add, add_smul, add_add_add_comm, smul_add