Theorem mul12 7293

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 7153	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
2	1	oveq1d 5552	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
3	2	3adant3 959	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( B x. A ) x. C ) )
4		mulass 7155	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulass 7155	. . 3 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B x. A ) x. C ) = ( B x. ( A x. C ) ) )
6	5	3com12 1143	. 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 2122	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 102   /\ w3a 920   = wceq 1285   e. wcel 1434  (class class class)co 5537   CCcc 7030   x. cmul 7037

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-mulcom 7128  ax-mulass 7130

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-iota 4891  df-fv 4934  df-ov 5540

This theorem is referenced by:  mul12i  7310  mul12d  7316  mulreap  9878  dvdscmul  10356  dvdstr  10366