Theorem mulcomli 8149

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 8148	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2250	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   x. cmul 8000

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211  ax-mulcom 8096

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  nummul2c  9623  halfthird  9716  5recm6rec  9717  sq4e2t8  10854  cos2bnd  12266  dec5nprm  12932  karatsuba  12948  2exp6  12951  2exp8  12953  2exp11  12954  2exp16  12955  2lgslem3a  15766  2lgsoddprmlem3c  15782  2lgsoddprmlem3d  15783  ex-exp  16049  ex-fac  16050