Theorem mul31 7239

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 7102	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 5548	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 956	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 7104	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulcl 7100	. . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
6	5	ancoms 264	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
7	6	3adant1 956	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
8		simp1 938	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
9	7, 8	mulcomd 7140	. 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 2123	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 102   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   x. cmul 6986

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-mulcl 7074  ax-mulcom 7077  ax-mulass 7079

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  mul31d  7262