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Theorem mul32d 9038
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
muld.1  |-  ( ph  ->  A  e.  CC )
addcomd.2  |-  ( ph  ->  B  e.  CC )
addcand.3  |-  ( ph  ->  C  e.  CC )
Assertion
Ref Expression
mul32d  |-  ( ph  ->  ( ( A  x.  B )  x.  C
)  =  ( ( A  x.  C )  x.  B ) )

Proof of Theorem mul32d
StepHypRef Expression
1 muld.1 . 2  |-  ( ph  ->  A  e.  CC )
2 addcomd.2 . 2  |-  ( ph  ->  B  e.  CC )
3 addcand.3 . 2  |-  ( ph  ->  C  e.  CC )
4 mul32 8995 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4syl3anc 1182 1  |-  ( ph  ->  ( ( A  x.  B )  x.  C
)  =  ( ( A  x.  C )  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    x. cmul 8758
This theorem is referenced by:  conjmul  9493  modmul1  11018  binom3  11238  bernneq  11243  expmulnbnd  11249  discr  11254  bcm1k  11343  bcp1n  11344  reccn2  12086  binomlem  12303  tanadd  12463  eirrlem  12498  dvds2ln  12575  bezoutlem4  12736  nrginvrcnlem  18217  tchcphlem2  18682  radcnvlem1  19805  tanarg  19986  cxpeq  20113  quad2  20151  binom4  20162  dquartlem2  20164  dquart  20165  quart1lem  20167  dvatan  20247  log2cnv  20256  basellem8  20341  bcmono  20532  lgsquadlem1  20609  rplogsumlem1  20649  dchrisumlem2  20655  chpdifbndlem1  20718  selberg3lem1  20722  selberg4  20726  selberg3r  20734  pntrlog2bndlem2  20743  pntrlog2bndlem3  20744  pntrlog2bndlem5  20746  pntlemf  20770  pntlemo  20772  ostth2lem1  20783  ostth2lem3  20800  circum  24022  faclimlem9  24125  csbrn  26565  jm2.25  27195  jm2.27c  27203  stirlinglem3  27928  cevathlem1  27960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-mulcom 8817  ax-mulass 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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