Theorem mul32 8156

Description: Commutative/associative law. (Contributed by NM, 8-Oct-1999.)

Assertion

Ref	Expression
mul32	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Proof of Theorem mul32

Step	Hyp	Ref	Expression
1		mulcom 8008	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 5938	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 1017	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 8010	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulass 8010	. . 3 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A x. C ) x. B ) = ( A x. ( C x. B ) ) )
6	5	3com23 1211	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) x. B ) = ( A x. ( C x. B ) ) )
7	3, 4, 6	3eqtr4d 2239	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( ( A x. C ) x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   x. cmul 7884

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-mulcom 7980  ax-mulass 7982

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  mul4  8158  mul32i  8173  mul32d  8179  muldvds1  11981  2sqlem6  15361