Theorem mulcomli 8327

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 8326	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2259	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1402   ∈ wcel 2209  (class class class)co 6079  ℂcc 8171   · cmul 8178

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 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220  ax-mulcom 8274

This theorem depends on definitions:  df-bi 117  df-cleq 2231

This theorem is referenced by:  nummul2c  9809  halfthird  9902  5recm6rec  9903  sq4e2t8  11057  cos2bnd  12510  dec5nprm  13176  karatsuba  13192  2exp6  13195  2exp8  13197  2exp11  13198  2exp16  13199  log2ublem3  16068  log2ublog2  16069  2lgslem3a  16195  2lgsoddprmlem3c  16211  2lgsoddprmlem3d  16212  ex-exp  16724  ex-fac  16725