Theorem mulcomli 7977

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 7976	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2208	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1363   ∈ wcel 2158  (class class class)co 5888  ℂcc 7822   · cmul 7829

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 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2169  ax-mulcom 7925

This theorem depends on definitions:  df-bi 117  df-cleq 2180

This theorem is referenced by:  nummul2c  9446  halfthird  9539  5recm6rec  9540  sq4e2t8  10631  cos2bnd  11781  2lgsoddprmlem3c  14728  2lgsoddprmlem3d  14729  ex-exp  14750  ex-fac  14751