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Theorem mul31 8980
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
StepHypRef Expression
1 mulcom 8823 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
21oveq2d 5874 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C )
)  =  ( A  x.  ( C  x.  B ) ) )
323adant1 973 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( A  x.  ( C  x.  B )
) )
4 mulass 8825 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulcl 8821 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
65ancoms 439 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
763adant1 973 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B )  e.  CC )
8 simp1 955 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
97, 8mulcomd 8856 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  B
)  x.  A )  =  ( A  x.  ( C  x.  B
) ) )
103, 4, 93eqtr4d 2325 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( C  x.  B )  x.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742
This theorem is referenced by:  mul02lem1  8988  addid1  8992  mul31d  9023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-mulcl 8799  ax-mulcom 8801  ax-mulass 8803
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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