-- Module: Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange
Goal:
R : Type u_1
A : Type u_2
V : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : AddCommGroup V
inst_3 : Algebra R A
inst_4 : Module R V
inst_5 : Invertible (α := R) 2
Q : QuadraticForm R V
this_1 : Invertible (α := A) 2 :=
  Invertible.copy (r := (algebraMap R A : (a : R) → A) 2) (Invertible.map (algebraMap R A) 2) 2 ⋯
this : Invertible (α := A ⊗[R] CliffordAlgebra Q) 2 :=
  Invertible.copy (r := (algebraMap R (A ⊗[R] CliffordAlgebra Q) : (a : R) → A ⊗[R] CliffordAlgebra Q) 2)
    (Invertible.map (algebraMap R (A ⊗[R] CliffordAlgebra Q)) 2 :
      Invertible (α := A ⊗[R] CliffordAlgebra Q)
        ((algebraMap R (A ⊗[R] CliffordAlgebra Q) : (a : R) → A ⊗[R] CliffordAlgebra Q) 2))
    2 ⋯
hpure_tensor :
  ∀ (v w : V),
    HAdd.hAdd (α := A ⊗[R] CliffordAlgebra Q)
        (HMul.hMul (α := A) 1 1 ⊗ₜ[R]
          HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
            ((ι Q : (a : V) → CliffordAlgebra Q) v : CliffordAlgebra Q)
            ((ι Q : (a : V) → CliffordAlgebra Q) w : CliffordAlgebra Q))
        (HMul.hMul (α := A) 1 1 ⊗ₜ[R]
          HMul.hMul (α := CliffordAlgebra Q) (β := CliffordAlgebra Q)
            ((ι Q : (a : V) → CliffordAlgebra Q) w : CliffordAlgebra Q)
            ((ι Q : (a : V) → CliffordAlgebra Q) v : CliffordAlgebra Q)) =
      ((((QuadraticMap.polarBilin (R := A) (M := A ⊗[R] V) (N := A) (QuadraticForm.baseChange A Q) :
                    LinearMap.BilinMap A (A ⊗[R] V) A) :
                  (a : A ⊗[R] V) → A ⊗[R] V →ₗ[A] A)
                (1 ⊗ₜ[R] v) :
              A ⊗[R] V →ₗ[A] A) :
            (a : A ⊗[R] V) → A)
          (1 ⊗ₜ[R] w) ⊗ₜ[R]
        1
rw :
  HAdd.hAdd (α := A ⊗[R] V →ₗ[A] A ⊗[R] V →ₗ[A] A ⊗[R] CliffordAlgebra Q)
      (LinearMap.compl₂ (LinearMap.mul A (A ⊗[R] CliffordAlgebra Q))
          (TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q)) ∘ₗ
        TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q))
      (LinearMap.compl₂ (LinearMap.flip (LinearMap.mul A (A ⊗[R] CliffordAlgebra Q)))
          (TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q)) ∘ₗ
        TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q)) =
    LinearMap.compr₂ (QuadraticMap.polarBilin (R := A) (M := A ⊗[R] V) (N := A) (QuadraticForm.baseChange A Q))
      (Algebra.linearMap A (A ⊗[R] CliffordAlgebra Q))
⊢ ∀ (m : A ⊗[R] V),
    HMul.hMul (α := A ⊗[R] CliffordAlgebra Q) (β := A ⊗[R] CliffordAlgebra Q)
        (((TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q) : A ⊗[R] V →ₗ[A] A ⊗[R] CliffordAlgebra Q) :
              (a : A ⊗[R] V) → A ⊗[R] CliffordAlgebra Q)
            m :
          A ⊗[R] CliffordAlgebra Q)
        (((TensorProduct.AlgebraTensorModule.map LinearMap.id (ι Q) : A ⊗[R] V →ₗ[A] A ⊗[R] CliffordAlgebra Q) :
              (a : A ⊗[R] V) → A ⊗[R] CliffordAlgebra Q)
            m :
          A ⊗[R] CliffordAlgebra Q) =
      (algebraMap A (A ⊗[R] CliffordAlgebra Q) : (a : A) → A ⊗[R] CliffordAlgebra Q)
        ((QuadraticForm.baseChange A Q : (a : A ⊗[R] V) → A) m)

Tactic:
rw [CliffordAlgebra.forall_mul_self_eq_iff (isUnit_of_invertible _)]

Premises used: isUnit_of_invertible, CliffordAlgebra.forall_mul_self_eq_iff