-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
Goal:
R : Type u1
inst : CommRing R
M : Type u2
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
d : Module.Dual R M
r : R
b : CliffordAlgebra Q
rw :
  (((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) (r • b) :
          Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
        (a : Module.Dual R M) → CliffordAlgebra Q)
      d =
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q)
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q)
rw₁ :
  HSMul.hSMul (β := CliffordAlgebra Q) r
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q) =
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q)
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q)
rw₂ rw₃ :
  HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q)
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q) =
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q)
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q)
⊢ (((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q)
            (HMul.hMul (α := CliffordAlgebra Q)
              ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q) b) :
          Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
        (a : Module.Dual R M) → CliffordAlgebra Q)
      d =
    HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r : CliffordAlgebra Q)
      ((((contractRight : (a : CliffordAlgebra Q) → Module.Dual R M →ₗ[R] CliffordAlgebra Q) b :
              Module.Dual R M →ₗ[R] CliffordAlgebra Q) :
            (a : Module.Dual R M) → CliffordAlgebra Q)
          d :
        CliffordAlgebra Q)

Tactic:
rw [← Algebra.smul_def, LinearMap.map_smul₂, Algebra.smul_def]

Premises used: LinearMap.map_smul₂, Algebra.smul_def