Theorem mulcomli 8281

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 8280	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2253	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   CCcc 8125   x. cmul 8132

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 2214  ax-mulcom 8228

This theorem depends on definitions:  df-bi 117  df-cleq 2225

This theorem is referenced by:  nummul2c  9758  halfthird  9851  5recm6rec  9852  sq4e2t8  10999  cos2bnd  12446  dec5nprm  13112  karatsuba  13128  2exp6  13131  2exp8  13133  2exp11  13134  2exp16  13135  2lgslem3a  15966  2lgsoddprmlem3c  15982  2lgsoddprmlem3d  15983  ex-exp  16495  ex-fac  16496