Theorem mulcomli 8026

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 8025	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2214	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1364   e. wcel 2164  (class class class)co 5918   CCcc 7870   x. cmul 7877

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 1458  ax-gen 1460  ax-4 1521  ax-17 1537  ax-ext 2175  ax-mulcom 7973

This theorem depends on definitions:  df-bi 117  df-cleq 2186

This theorem is referenced by:  nummul2c  9497  halfthird  9590  5recm6rec  9591  sq4e2t8  10708  cos2bnd  11903  2lgsoddprmlem3c  15197  2lgsoddprmlem3d  15198  ex-exp  15219  ex-fac  15220