Theorem mul12 7884

Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)

Assertion

Ref	Expression
mul12	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

Proof of Theorem mul12

Step	Hyp	Ref	Expression
1		mulcom 7742	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2	1	oveq1d 5782	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
3	2	3adant3 1001	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
4		mulass 7744	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulass 7744	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) )
6	5	3com12 1185	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) )
7	3, 4, 6	3eqtr3d 2178	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( B x. ( A x. C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5767   CCcc 7611   x. cmul 7618

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-mulcom 7714  ax-mulass 7716

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-iota 5083  df-fv 5126  df-ov 5770

This theorem is referenced by:  mul12i  7901  mul12d  7907  mulreap  10629  demoivre  11468  demoivreALT  11469  dvdscmul  11509  dvdstr  11519  sinperlem  12878  coskpi  12918