Theorem mulcomli 7906

Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	⊢ 𝐴 ∈ ℂ
axi.2	⊢ 𝐵 ∈ ℂ
mulcomli.3	⊢ (𝐴 · 𝐵) = 𝐶

Assertion

Ref	Expression
mulcomli	⊢ (𝐵 · 𝐴) = 𝐶

Proof of Theorem mulcomli

Step	Hyp	Ref	Expression
1		axi.2	. . 3 ⊢ 𝐵 ∈ ℂ
2		axi.1	. . 3 ⊢ 𝐴 ∈ ℂ
3	1, 2	mulcomi 7905	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2186	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1343   ∈ wcel 2136  (class class class)co 5842  ℂcc 7751   · cmul 7758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147  ax-mulcom 7854

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  nummul2c  9371  halfthird  9464  5recm6rec  9465  sq4e2t8  10552  cos2bnd  11701  ex-exp  13608  ex-fac  13609