Theorem mulcomli 8191

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 8190	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2251	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1397   ∈ wcel 2201  (class class class)co 6023  ℂcc 8035   · cmul 8042

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 1495  ax-gen 1497  ax-4 1558  ax-17 1574  ax-ext 2212  ax-mulcom 8138

This theorem depends on definitions:  df-bi 117  df-cleq 2223

This theorem is referenced by:  nummul2c  9665  halfthird  9758  5recm6rec  9759  sq4e2t8  10905  cos2bnd  12344  dec5nprm  13010  karatsuba  13026  2exp6  13029  2exp8  13031  2exp11  13032  2exp16  13033  2lgslem3a  15851  2lgsoddprmlem3c  15867  2lgsoddprmlem3d  15868  ex-exp  16380  ex-fac  16381