-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv
Goal:
R : Type u_1
M : Type u_2
inst : CommRing R
inst_1 : AddCommGroup M
inst_2 : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
  (AlgHom.toLinearMap (Algebra.lmul R (CliffordAlgebra Q)) ∘ₗ
        (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)).compl₂
    (ι Q ∘ₗ LinearMap.fst R M R - Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)
f_apply :
  ∀ (x y : M × R),
    (((f : (a : M × R) → M × R →ₗ[R] CliffordAlgebra Q) x : M × R →ₗ[R] CliffordAlgebra Q) :
          (a : M × R) → CliffordAlgebra Q)
        y =
      HAdd.hAdd (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) x.1 : CliffordAlgebra Q)
          ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) x.2 : CliffordAlgebra Q) *
        HSub.hSub (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
          ((ι Q : (a : M) → CliffordAlgebra Q) y.1 : CliffordAlgebra Q)
          ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) y.2 : CliffordAlgebra Q)
hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) r) x
m : M × R
rw :
  HSub.hSub (α := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q : (a : M) → R) m.1) : CliffordAlgebra Q)
      (HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) m.2 : CliffordAlgebra Q)
        ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) m.2 : CliffordAlgebra Q)) =
    (algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q' Q : (a : M × R) → R) m)
rw₁ :
  HSub.hSub (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q : (a : M) → R) m.1) : CliffordAlgebra Q)
      ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) (m.2 * m.2) : CliffordAlgebra Q) =
    (algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q' Q : (a : M × R) → R) m)
rw₂ :
  Eq (α := CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HSub.hSub (α := R) ((Q : (a : M) → R) m.1 : R) (m.2 * m.2)) :
      CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q' Q : (a : M × R) → R) m) : CliffordAlgebra Q)
rw₃ :
  Eq (α := CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HAdd.hAdd (α := R) ((Q : (a : M) → R) m.1 : R) (-(m.2 * m.2))) :
      CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q' Q : (a : M × R) → R) m) : CliffordAlgebra Q)
rw₄ :
  Eq (α := CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HAdd.hAdd (α := R) ((Q : (a : M) → R) m.1 : R) (-(m.2 * m.2))) :
      CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HSub.hSub (α := R) ((Q : (a : M) → R) m.1 : R) (m.2 * m.2)) :
      CliffordAlgebra Q)
rw₅ rw₆ :
  Eq (α := CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HAdd.hAdd (α := R) ((Q : (a : M) → R) m.1 : R) (-(m.2 * m.2))) :
      CliffordAlgebra Q)
    ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q)
        (HAdd.hAdd (α := R) ((Q : (a : M) → R) m.1 : R) (-(m.2 * m.2))) :
      CliffordAlgebra Q)
⊢ HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m.1 : CliffordAlgebra Q)
        ((ι Q : (a : M) → CliffordAlgebra Q) m.1 : CliffordAlgebra Q) -
      HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
        ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) m.2 : CliffordAlgebra Q)
        ((algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) m.2 : CliffordAlgebra Q) =
    (algebraMap R (CliffordAlgebra Q) : (a : R) → CliffordAlgebra Q) ((Q' Q : (a : M × R) → R) m)

Tactic:
rw [ι_sq_scalar, ← map_mul, ← map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg]

Premises used: map_sub, CliffordAlgebra.ι_sq_scalar, sub_eq_add_neg, CliffordAlgebra.EquivEven.Q'_apply, map_mul