Theorem mulcomli 8179

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 8178	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2250	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1395   ∈ wcel 2200  (class class class)co 6013  ℂcc 8023   · cmul 8030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211  ax-mulcom 8126

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  nummul2c  9653  halfthird  9746  5recm6rec  9747  sq4e2t8  10892  cos2bnd  12314  dec5nprm  12980  karatsuba  12996  2exp6  12999  2exp8  13001  2exp11  13002  2exp16  13003  2lgslem3a  15815  2lgsoddprmlem3c  15831  2lgsoddprmlem3d  15832  ex-exp  16273  ex-fac  16274