Theorem mulcomli 8185

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 8184	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2252	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1397   e. wcel 2202  (class class class)co 6017   CCcc 8029   x. cmul 8036

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 2213  ax-mulcom 8132

This theorem depends on definitions:  df-bi 117  df-cleq 2224

This theorem is referenced by:  nummul2c  9659  halfthird  9752  5recm6rec  9753  sq4e2t8  10898  cos2bnd  12320  dec5nprm  12986  karatsuba  13002  2exp6  13005  2exp8  13007  2exp11  13008  2exp16  13009  2lgslem3a  15821  2lgsoddprmlem3c  15837  2lgsoddprmlem3d  15838  ex-exp  16323  ex-fac  16324