Theorem mulcomli 7776

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 7775	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2160	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7621   x. cmul 7628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506  ax-ext 2121  ax-mulcom 7724

This theorem depends on definitions:  df-bi 116  df-cleq 2132

This theorem is referenced by:  nummul2c  9234  halfthird  9327  5recm6rec  9328  sq4e2t8  10393  cos2bnd  11470  ex-exp  12942  ex-fac  12943