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Theorem mul32d 9591
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 9548 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4syl3anc 1218 1  |-  ( ph  ->  ( ( A  x.  B )  x.  C
)  =  ( ( A  x.  C )  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756  (class class class)co 6103   CCcc 9292    x. cmul 9299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-mulcom 9358  ax-mulass 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-ov 6106
This theorem is referenced by:  conjmul  10060  modmul1  11764  binom3  11997  bernneq  12002  expmulnbnd  12008  discr  12013  bcm1k  12103  bcp1n  12104  reccn2  13086  binomlem  13304  tanadd  13463  eirrlem  13498  dvds2ln  13575  bezoutlem4  13737  modprm0  13885  nrginvrcnlem  20283  tchcphlem2  20763  csbren  20910  radcnvlem1  21890  tanarg  22080  cxpeq  22207  quad2  22246  binom4  22257  dquartlem2  22259  dquart  22260  quart1lem  22262  dvatan  22342  log2cnv  22351  basellem8  22437  bcmono  22628  lgsquadlem1  22705  rplogsumlem1  22745  dchrisumlem2  22751  chpdifbndlem1  22814  selberg3lem1  22818  selberg4  22822  selberg3r  22830  pntrlog2bndlem2  22839  pntrlog2bndlem3  22840  pntrlog2bndlem5  22842  pntlemf  22866  pntlemo  22868  ostth2lem1  22879  ostth2lem3  22896  circum  27331  binomfallfaclem2  27555  jm2.25  29360  jm2.27c  29368  stirlinglem3  29883  cevathlem1  29915
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