Theorem mulcomli 8229

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 8228	. 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 1398   e. wcel 2202  (class class class)co 6028   CCcc 8073   x. cmul 8080

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 2213  ax-mulcom 8176

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

This theorem is referenced by:  nummul2c  9704  halfthird  9797  5recm6rec  9798  sq4e2t8  10945  cos2bnd  12384  dec5nprm  13050  karatsuba  13066  2exp6  13069  2exp8  13071  2exp11  13072  2exp16  13073  2lgslem3a  15895  2lgsoddprmlem3c  15911  2lgsoddprmlem3d  15912  ex-exp  16424  ex-fac  16425