-- Mathlib/LinearAlgebra/TensorPower/Basic.lean
theorem mul_assoc {na nb nc} (a : (⨂[R]^na) M) (b : (⨂[R]^nb) M) (c : (⨂[R]^nc) M) :
    cast R M (add_assoc _ _ _) (a ₜ* b ₜ* c) = a ₜ* (b ₜ* c)

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a : ⨂[R]^na M
b : ⨂[R]^nb M
c : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
lhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^(na + nb) M) (⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
      ((mul (na + nb) nc).compr₂ ↑e) ∘ₗ
    mul na nb
lhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M),
    ((lhs a) b) c = e (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c)
rhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (((LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^nc M →ₗ[R] ⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M)
          (⨂[R]^na M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
        LinearMap.lflip)
      ((LinearMap.llcomp R (⨂[R]^nc M) (⨂[R]^(nb + nc) M) (⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
          (mul na (nb + nc)).flip ∘ₗ
        mul nb nc)).flip
rhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M), ((rhs a) b) c = GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)
⊢ (cast R M ⋯) (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c) =
    GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)

Tactic:
suffices lhs = rhs from
  LinearMap.congr_fun (LinearMap.congr_fun (LinearMap.congr_fun this a) b) c

Result:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a : ⨂[R]^na M
b : ⨂[R]^nb M
c : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
lhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^(na + nb) M) (⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
      ((mul (na + nb) nc).compr₂ ↑e) ∘ₗ
    mul na nb
lhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M),
    ((lhs a) b) c = e (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c)
rhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (((LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^nc M →ₗ[R] ⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M)
          (⨂[R]^na M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
        LinearMap.lflip)
      ((LinearMap.llcomp R (⨂[R]^nc M) (⨂[R]^(nb + nc) M) (⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
          (mul na (nb + nc)).flip ∘ₗ
        mul nb nc)).flip
rhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M), ((rhs a) b) c = GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)
⊢ lhs = rhs

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a : ⨂[R]^na M
b : ⨂[R]^nb M
c : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
lhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^(na + nb) M) (⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
      ((mul (na + nb) nc).compr₂ ↑e) ∘ₗ
    mul na nb
lhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M),
    ((lhs a) b) c = e (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c)
rhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (((LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^nc M →ₗ[R] ⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M)
          (⨂[R]^na M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
        LinearMap.lflip)
      ((LinearMap.llcomp R (⨂[R]^nc M) (⨂[R]^(nb + nc) M) (⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
          (mul na (nb + nc)).flip ∘ₗ
        mul nb nc)).flip
rhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M), ((rhs a) b) c = GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)
⊢ lhs = rhs

Tactic:
ext a b c

Result:
case H.H.H.H.H.H
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a✝ : ⨂[R]^na M
b✝ : ⨂[R]^nb M
c✝ : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
lhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^(na + nb) M) (⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
      ((mul (na + nb) nc).compr₂ ↑e) ∘ₗ
    mul na nb
lhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M),
    ((lhs a) b) c = e (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c)
rhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (((LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^nc M →ₗ[R] ⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M)
          (⨂[R]^na M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
        LinearMap.lflip)
      ((LinearMap.llcomp R (⨂[R]^nc M) (⨂[R]^(nb + nc) M) (⨂[R]^na M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
          (mul na (nb + nc)).flip ∘ₗ
        mul nb nc)).flip
rhs_eq :
  ∀ (a : ⨂[R]^na M) (b : ⨂[R]^nb M) (c : ⨂[R]^nc M), ((rhs a) b) c = GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)
a : Fin na → M
b : Fin nb → M
c : Fin nc → M
⊢ (((((lhs.compMultilinearMap (tprod R)) a).compMultilinearMap (tprod R)) b).compMultilinearMap (tprod R)) c =
    (((((rhs.compMultilinearMap (tprod R)) a).compMultilinearMap (tprod R)) b).compMultilinearMap (tprod R)) c

Proof state:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a : ⨂[R]^na M
b : ⨂[R]^nb M
c : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
⊢ (cast R M ⋯) (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c) =
    GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)

Tactic:
let lhs : (⨂[R]^na) M →ₗ[R] (⨂[R]^nb) M →ₗ[R] (⨂[R]^nc) M →ₗ[R] (⨂[R]^(na + (nb + nc))) M :=
  (LinearMap.llcomp R _ _ _ ((mul _ nc).compr₂ e.toLinearMap)).comp (mul na nb)

Result:
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
na nb nc : ℕ
a : ⨂[R]^na M
b : ⨂[R]^nb M
c : ⨂[R]^nc M
mul : (n m : ℕ) → ⨂[R]^n M →ₗ[R] ⨂[R]^m M →ₗ[R] ⨂[R]^(n + m) M :=
  fun n m => (TensorProduct.mk R (⨂[R]^n M) (⨂[R]^m M)).compr₂ ↑mulEquiv
e : ⨂[R]^(na + nb + nc) M ≃ₗ[R] ⨂[R]^(na + (nb + nc)) M := cast R M ⋯
lhs : ⨂[R]^na M →ₗ[R] ⨂[R]^nb M →ₗ[R] ⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M :=
  (LinearMap.llcomp R (⨂[R]^nb M) (⨂[R]^(na + nb) M) (⨂[R]^nc M →ₗ[R] ⨂[R]^(na + (nb + nc)) M))
      ((mul (na + nb) nc).compr₂ ↑e) ∘ₗ
    mul na nb
⊢ (cast R M ⋯) (GradedMonoid.GMul.mul (GradedMonoid.GMul.mul a b) c) =
    GradedMonoid.GMul.mul a (GradedMonoid.GMul.mul b c)