Theorem mul12 8027

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 7882	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2	1	oveq1d 5857	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
3	2	3adant3 1007	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
4		mulass 7884	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulass 7884	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) )
6	5	3com12 1197	. 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 2206	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 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   x. cmul 7758

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-mulcom 7854  ax-mulass 7856

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  mul12i  8044  mul12d  8050  mulreap  10806  demoivre  11713  demoivreALT  11714  dvdscmul  11758  dvdstr  11768  sinperlem  13369  coskpi  13409