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Theorem mul32d 9793
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 9750 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4syl3anc 1229 1  |-  ( ph  ->  ( ( A  x.  B )  x.  C
)  =  ( ( A  x.  C )  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383    e. wcel 1804  (class class class)co 6281   CCcc 9493    x. cmul 9500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-mulcom 9559  ax-mulass 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284
This theorem is referenced by:  conjmul  10268  modmul1  12022  binom3  12269  bernneq  12274  expmulnbnd  12280  discr  12285  bcm1k  12375  bcp1n  12376  reccn2  13401  binomlem  13623  tanadd  13884  eirrlem  13919  dvds2ln  13996  bezoutlem4  14161  modprm0  14312  nrginvrcnlem  21177  tchcphlem2  21657  csbren  21804  radcnvlem1  22786  tanarg  22982  cxpeq  23109  quad2  23148  binom4  23159  dquartlem2  23161  dquart  23162  quart1lem  23164  dvatan  23244  log2cnv  23253  basellem8  23339  bcmono  23530  lgsquadlem1  23607  rplogsumlem1  23647  dchrisumlem2  23653  chpdifbndlem1  23716  selberg3lem1  23720  selberg4  23724  selberg3r  23732  pntrlog2bndlem2  23741  pntrlog2bndlem3  23742  pntrlog2bndlem5  23744  pntlemf  23768  pntlemo  23770  ostth2lem1  23781  ostth2lem3  23798  circum  29018  binomfallfaclem2  29138  jm2.25  30917  jm2.27c  30925  dvasinbx  31671  stirlinglem3  31812  dirkercncflem2  31840  cevathlem1  32038
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