Theorem mul31 8090

Description: Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)

Assertion

Ref	Expression
mul31	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )

Proof of Theorem mul31

Step	Hyp	Ref	Expression
1		mulcom 7942	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 5893	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 1015	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 7944	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulcl 7940	. . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
6	5	ancoms 268	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
7	6	3adant1 1015	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
8		simp1 997	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
9	7, 8	mulcomd 7981	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C x. B ) x. A ) = ( A x. ( C x. B ) ) )
10	3, 4, 9	3eqtr4d 2220	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( C x. B ) x. A ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5877   CCcc 7811   x. cmul 7818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-mulcl 7911  ax-mulcom 7914  ax-mulass 7916

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880

This theorem is referenced by:  mul31d  8113