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Theorem mul12 9237
Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
mul12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )

Proof of Theorem mul12
StepHypRef Expression
1 mulcom 9081 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6099 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  C
)  =  ( ( B  x.  A )  x.  C ) )
323adant3 978 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( B  x.  A )  x.  C ) )
4 mulass 9083 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulass 9083 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
653com12 1158 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
73, 4, 63eqtr3d 2478 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726  (class class class)co 6084   CCcc 8993    x. cmul 9000
This theorem is referenced by:  mul02  9249  mul12i  9266  mul12d  9280  mulre  11931  sqreulem  12168  demoivre  12806  demoivreALT  12807  dvdscmul  12881  dvdscmulr  12883  dvdstr  12888  ablfacrp  15629  nmoleub2lem3  19128  sinperlem  20393  coskpi  20433  sineq0  20434  efif1olem4  20452  rpvmasum2  21211  fsumcube  26111  expgrowthi  27541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-mulcom 9059  ax-mulass 9061
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-ov 6087
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