-- Lean 4 proof step from Mathlib.LinearAlgebra.CliffordAlgebra.Prod (goal)
Goal:
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst : CommRing R
inst_1 : AddCommGroup M₁
inst_2 : AddCommGroup M₂
inst_3 : AddCommGroup N
inst_4 : Module R M₁
inst_5 : Module R M₂
inst_6 : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ ((f₁ : (a : M₁) → N) x) ((f₂ : (a : M₂) → N) y)
m₁ m₁' : CliffordAlgebra Q₁
i₁n i : ℕ
x₁ : CliffordAlgebra Q₁
_hx₁ : x₁ ∈ (ι Q₁).range ^ i
v₁ : M₁
m₂' : CliffordAlgebra Q₂
i₂n : ℕ
r : R
mem :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
mem₁ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) 0 •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
mem₂ :
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
    HSMul.hSMul (α := ℤˣ) 1
      (HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ))
⊢ HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
      ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
      ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
          ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
        CliffordAlgebra Qₙ) =
    HPow.hPow (α := ℤˣ) (β := ZMod 2) (-1) (↑0 : ZMod 2) •
      HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
        ((map f₂ : (a : CliffordAlgebra Q₂) → CliffordAlgebra Qₙ)
            ((algebraMap R (CliffordAlgebra Q₂) : (a : R) → CliffordAlgebra Q₂) r) :
          CliffordAlgebra Qₙ)
        ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
have mem₃ : HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
    ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ) =
  HMul.hMul (α := CliffordAlgebra Qₙ) (β := CliffordAlgebra Qₙ)
    ((algebraMap R (CliffordAlgebra Qₙ) : (a : R) → CliffordAlgebra Qₙ) r : CliffordAlgebra Qₙ)
    ((ι Qₙ : (a : N) → CliffordAlgebra Qₙ) ((f₁ : (a : M₁) → N) v₁) : CliffordAlgebra Qₙ)
Tactic: rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]