Theorem mul31 8447

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 8298	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 6091	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 1046	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 8300	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulcl 8296	. . . . 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 1046	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
8		simp1 1028	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
9	7, 8	mulcomd 8337	. 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 2281	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 1009   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   x. cmul 8174

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-mulcl 8267  ax-mulcom 8270  ax-mulass 8272

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  mul31d  8470