Theorem mulcomli 8297

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 8296	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2255	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1398   ∈ wcel 2205  (class class class)co 6058  ℂcc 8141   · cmul 8148

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216  ax-mulcom 8244

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  nummul2c  9779  halfthird  9872  5recm6rec  9873  sq4e2t8  11026  cos2bnd  12474  dec5nprm  13140  karatsuba  13156  2exp6  13159  2exp8  13161  2exp11  13162  2exp16  13163  2lgslem3a  16095  2lgsoddprmlem3c  16111  2lgsoddprmlem3d  16112  ex-exp  16624  ex-fac  16625