Theorem mulcomli 8099

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 8098	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2227	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1373   e. wcel 2177  (class class class)co 5957   CCcc 7943   x. cmul 7950

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2188  ax-mulcom 8046

This theorem depends on definitions:  df-bi 117  df-cleq 2199

This theorem is referenced by:  nummul2c  9573  halfthird  9666  5recm6rec  9667  sq4e2t8  10804  cos2bnd  12146  dec5nprm  12812  karatsuba  12828  2exp6  12831  2exp8  12833  2exp11  12834  2exp16  12835  2lgslem3a  15645  2lgsoddprmlem3c  15661  2lgsoddprmlem3d  15662  ex-exp  15802  ex-fac  15803