Theorem mulcomli 7964

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 7963	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2198	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   x. cmul 7816

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 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159  ax-mulcom 7912

This theorem depends on definitions:  df-bi 117  df-cleq 2170

This theorem is referenced by:  nummul2c  9433  halfthird  9526  5recm6rec  9527  sq4e2t8  10618  cos2bnd  11768  2lgsoddprmlem3c  14460  2lgsoddprmlem3d  14461  ex-exp  14482  ex-fac  14483