Theorem mulcomli 8323

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 8322	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2259	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   x. cmul 8174

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 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220  ax-mulcom 8270

This theorem depends on definitions:  df-bi 117  df-cleq 2231

This theorem is referenced by:  nummul2c  9805  halfthird  9898  5recm6rec  9899  sq4e2t8  11052  cos2bnd  12505  dec5nprm  13171  karatsuba  13187  2exp6  13190  2exp8  13192  2exp11  13193  2exp16  13194  2lgslem3a  16126  2lgsoddprmlem3c  16142  2lgsoddprmlem3d  16143  ex-exp  16655  ex-fac  16656