Theorem mul31 7893

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 7749	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 5790	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 999	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 7751	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulcl 7747	. . . . 5 |- ( ( C e. CC /\ B e. CC ) -> ( C x. B ) e. CC )
6	5	ancoms 266	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
7	6	3adant1 999	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
8		simp1 981	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
9	7, 8	mulcomd 7787	. 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 2182	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 103   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   x. cmul 7625

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 2121  ax-mulcl 7718  ax-mulcom 7721  ax-mulass 7723

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  mul31d  7916