Theorem mulcomli 8086

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 8085	. 2 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵)
4		mulcomli.3	. 2 ⊢ (𝐴 · 𝐵) = 𝐶
5	3, 4	eqtri 2227	1 ⊢ (𝐵 · 𝐴) = 𝐶

Syntax hints:   = wceq 1373   ∈ wcel 2177  (class class class)co 5951  ℂcc 7930   · cmul 7937

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188  ax-mulcom 8033

This theorem depends on definitions:  df-bi 117  df-cleq 2199

This theorem is referenced by:  nummul2c  9560  halfthird  9653  5recm6rec  9654  sq4e2t8  10789  cos2bnd  12115  dec5nprm  12781  karatsuba  12797  2exp6  12800  2exp8  12802  2exp11  12803  2exp16  12804  2lgslem3a  15614  2lgsoddprmlem3c  15630  2lgsoddprmlem3d  15631  ex-exp  15737  ex-fac  15738