-- Mathlib/LinearAlgebra/Basis/Basic.lean
/-- An element of a non-unital-non-associative algebra is in the center exactly when it commutes
with the basis elements. -/
lemma Basis.mem_center_iff {A}
    [Semiring R] [NonUnitalNonAssocSemiring A]
    [Module R A] [SMulCommClass R A A] [SMulCommClass R R A] [IsScalarTower R A A]
    (b : Basis ι R A) {z : A} :
    z ∈ Set.center A ↔
      (∀ i, Commute (b i) z) ∧ ∀ i j,
        z * (b i * b j) = (z * b i) * b j
          ∧ (b i * z) * b j = b i * (z * b j)
          ∧ (b i * b j) * z = b i * (b j * z)

Proof state:
case mpr.left_assoc
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
⊢ ∀ (b c : A), z * (b * c) = z * b * c

case mpr.mid_assoc
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
⊢ ∀ (a c : A), a * z * c = a * (z * c)

case mpr.right_assoc
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
⊢ ∀ (a b : A), a * b * z = a * (b * z)

Tactic:
case left_assoc =>
  intro c d
  rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
      linearCombination_apply, sum, sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum,
      Finset.mul_sum]
  simp_rw [smul_mul_assoc, Finset.mul_sum, Finset.sum_mul, mul_smul_comm, Finset.mul_sum,
    Finset.smul_sum, smul_mul_assoc, mul_smul_comm, (h.2 _ _).1,
    (@SMulCommClass.smul_comm R R A)]
  rw [Finset.sum_comm]

Result:
case mpr.mid_assoc
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
⊢ ∀ (a c : A), a * z * c = a * (z * c)

case mpr.right_assoc
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
⊢ ∀ (a b : A), a * b * z = a * (b * z)

Proof state:
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
c d : A
⊢ c * d * z = c * (d * z)

Tactic:
rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
    linearCombination_apply, sum, Finsupp.sum, Finset.sum_mul]

Result:
ι : Type u_1
R : Type u_3
A : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring A
inst✝³ : Module R A
inst✝² : SMulCommClass R A A
inst✝¹ : SMulCommClass R R A
inst✝ : IsScalarTower R A A
b : Basis ι R A
z : A
h :
  (∀ (i : ι), Commute (b i) z) ∧
    ∀ (i j : ι), z * (b i * b j) = z * b i * b j ∧ b i * z * b j = b i * (z * b j) ∧ b i * b j * z = b i * (b j * z)
c d : A
⊢ (∑ i ∈ (b.repr c).support, (b.repr c) i • b i * ∑ a ∈ (b.repr d).support, (b.repr d) a • b a) * z =
    (∑ a ∈ (b.repr c).support, (b.repr c) a • b a) * ((∑ a ∈ (b.repr d).support, (b.repr d) a • b a) * z)