Theorem mulcomli 8297

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 8296	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2255	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   x. cmul 8148

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 1496  ax-gen 1498  ax-4 1559  ax-17 1575  ax-ext 2216  ax-mulcom 8244

This theorem depends on definitions:  df-bi 117  df-cleq 2227

This theorem is referenced by:  nummul2c  9776  halfthird  9869  5recm6rec  9870  sq4e2t8  11023  cos2bnd  12471  dec5nprm  13137  karatsuba  13153  2exp6  13156  2exp8  13158  2exp11  13159  2exp16  13160  2lgslem3a  16092  2lgsoddprmlem3c  16108  2lgsoddprmlem3d  16109  ex-exp  16621  ex-fac  16622