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Theorem mul31 9624
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 9455 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
21oveq2d 6192 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C )
)  =  ( A  x.  ( C  x.  B ) ) )
323adant1 1006 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( A  x.  ( C  x.  B )
) )
4 mulass 9457 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulcl 9453 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
65ancoms 453 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
763adant1 1006 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B )  e.  CC )
8 simp1 988 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
97, 8mulcomd 9494 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  B
)  x.  A )  =  ( A  x.  ( C  x.  B
) ) )
103, 4, 93eqtr4d 2500 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( C  x.  B )  x.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1757  (class class class)co 6176   CCcc 9367    x. cmul 9374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-mulcl 9431  ax-mulcom 9433  ax-mulass 9435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-rex 2798  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-br 4377  df-iota 5465  df-fv 5510  df-ov 6179
This theorem is referenced by:  mul02lem1  9632  addid1  9636  mul31d  9667
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