-- Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean
theorem toBaseChange_reverse (Q : QuadraticForm R V) (x : CliffordAlgebra (Q.baseChange A)) :
    toBaseChange A Q (reverse x) =
      TensorProduct.map LinearMap.id reverse (toBaseChange A Q x)

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A).symm ∘ₗ ↑(opLinearEquiv A))
        (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      (↑(toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)

Tactic:
dsimp

Result:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A ≪≫ₗ (opLinearEquiv A).symm))
        (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      ((toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ unop
      (((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
          ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
        x) =
    (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)

Tactic:
refine (LinearMap.congr_fun (TensorProduct.AlgebraTensorModule.map_comp _ _ _ _).symm _).trans ?_

Result:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A).symm ∘ₗ (↑(AlgEquiv.toOpposite A A)).toLinearMap)
        (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      (↑(toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(LinearEquiv.refl A A)) (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      ((toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)

Tactic:
rfl

Result:
no goals

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
this :
  ((AlgHom.op (toBaseChange A Q)).comp reverseOp) x =
    ((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
        ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
      x
⊢ unop
      (((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
          ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
        x) =
    (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)

Tactic:
clear this

Result:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ unop
      (((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
          ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
        x) =
    (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A ≪≫ₗ (opLinearEquiv A).symm))
        (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      ((toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)

Tactic:
rw [LinearEquiv.self_trans_symm]

Result:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(LinearEquiv.refl A A)) (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
      ((toBaseChange A Q) x) =
    (TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)

Proof state:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
this :
  ((AlgHom.op (toBaseChange A Q)).comp reverseOp) x =
    ((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
        ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
      x
⊢ (toBaseChange A Q) (reverse x) = (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)

Tactic:
refine (congr_arg unop this).trans ?_

Result:
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
this :
  ((AlgHom.op (toBaseChange A Q)).comp reverseOp) x =
    ((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
        ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
      x
⊢ unop
      (((↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
          ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q)))
        x) =
    (TensorProduct.map LinearMap.id reverse) ((toBaseChange A Q) x)