Theorem mulcomli 8033

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 8032	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2217	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   x. cmul 7884

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178  ax-mulcom 7980

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  nummul2c  9506  halfthird  9599  5recm6rec  9600  sq4e2t8  10729  cos2bnd  11925  dec5nprm  12583  karatsuba  12599  2exp6  12602  2exp8  12604  2exp11  12605  2exp16  12606  2lgslem3a  15334  2lgsoddprmlem3c  15350  2lgsoddprmlem3d  15351  ex-exp  15373  ex-fac  15374