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Theorem mul12 9196
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 9040 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6063 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  C
)  =  ( ( B  x.  A )  x.  C ) )
323adant3 977 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( B  x.  A )  x.  C ) )
4 mulass 9042 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulass 9042 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
653com12 1157 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
73, 4, 63eqtr3d 2452 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721  (class class class)co 6048   CCcc 8952    x. cmul 8959
This theorem is referenced by:  mul02  9208  mul12i  9225  mul12d  9239  mulre  11889  sqreulem  12126  demoivre  12764  demoivreALT  12765  dvdscmul  12839  dvdscmulr  12841  dvdstr  12846  ablfacrp  15587  nmoleub2lem3  19084  sinperlem  20349  coskpi  20389  sineq0  20390  efif1olem4  20408  rpvmasum2  21167  fsumcube  26018  expgrowthi  27426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-mulcom 9018  ax-mulass 9020
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-ov 6051
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