Theorem mulcomli 8121

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 8120	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2230	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1375   ∈ wcel 2180  (class class class)co 5974  ℂcc 7965   · cmul 7972

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 1473  ax-gen 1475  ax-4 1536  ax-17 1552  ax-ext 2191  ax-mulcom 8068

This theorem depends on definitions:  df-bi 117  df-cleq 2202

This theorem is referenced by:  nummul2c  9595  halfthird  9688  5recm6rec  9689  sq4e2t8  10826  cos2bnd  12237  dec5nprm  12903  karatsuba  12919  2exp6  12922  2exp8  12924  2exp11  12925  2exp16  12926  2lgslem3a  15737  2lgsoddprmlem3c  15753  2lgsoddprmlem3d  15754  ex-exp  16001  ex-fac  16002