Theorem mulcomli 8061

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 8060	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2225	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1372   e. wcel 2175  (class class class)co 5934   CCcc 7905   x. cmul 7912

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 1469  ax-gen 1471  ax-4 1532  ax-17 1548  ax-ext 2186  ax-mulcom 8008

This theorem depends on definitions:  df-bi 117  df-cleq 2197

This theorem is referenced by:  nummul2c  9535  halfthird  9628  5recm6rec  9629  sq4e2t8  10763  cos2bnd  11990  dec5nprm  12656  karatsuba  12672  2exp6  12675  2exp8  12677  2exp11  12678  2exp16  12679  2lgslem3a  15488  2lgsoddprmlem3c  15504  2lgsoddprmlem3d  15505  ex-exp  15527  ex-fac  15528