Theorem mulcomli 7558

Description: Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)

Hypotheses

Ref	Expression
axi.1	|- A e. CC
axi.2	|- B e. CC
mulcomli.3	|- ( A x. B ) = C

Assertion

Ref	Expression
mulcomli	|- ( B x. A ) = C

Proof of Theorem mulcomli

Step	Hyp	Ref	Expression
1		axi.2	. . 3 |- B e. CC
2		axi.1	. . 3 |- A e. CC
3	1, 2	mulcomi 7557	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2109	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1290   e. wcel 1439  (class class class)co 5668   CCcc 7411   x. cmul 7418

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071  ax-mulcom 7509

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  nummul2c  8989  sq4e2t8  10115  cos2bnd  11114  ex-exp  11958  ex-fac  11959