Theorem mulcomli 7886

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 7885	. 2 |- ( B x. A ) = ( A x. B )
4		mulcomli.3	. 2 |- ( A x. B ) = C
5	3, 4	eqtri 2178	1 |- ( B x. A ) = C

Syntax hints:   = wceq 1335   e. wcel 2128  (class class class)co 5825   CCcc 7731   x. cmul 7738

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 1427  ax-gen 1429  ax-4 1490  ax-17 1506  ax-ext 2139  ax-mulcom 7834

This theorem depends on definitions:  df-bi 116  df-cleq 2150

This theorem is referenced by:  nummul2c  9345  halfthird  9438  5recm6rec  9439  sq4e2t8  10520  cos2bnd  11661  ex-exp  13345  ex-fac  13346