Theorem mul31 8218

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 8069	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
2	1	oveq2d 5972	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
3	2	3adant1 1018	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B x. C ) ) = ( A x. ( C x. B ) ) )
4		mulass 8071	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. B ) x. C ) = ( A x. ( B x. C ) ) )
5		mulcl 8067	. . . . 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 1018	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. B ) e. CC )
8		simp1 1000	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
9	7, 8	mulcomd 8109	. 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 2249	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 981   = wceq 1373   e. wcel 2177  (class class class)co 5956   CCcc 7938   x. cmul 7945

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-mulcl 8038  ax-mulcom 8041  ax-mulass 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-iota 5240  df-fv 5287  df-ov 5959

This theorem is referenced by:  mul31d  8241