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Theorem mul12d 9021
Description: Commutative/associative law that swaps the first 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
mul12d  |-  ( ph  ->  ( A  x.  ( B  x.  C )
)  =  ( B  x.  ( A  x.  C ) ) )

Proof of Theorem mul12d
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 mul12 8978 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
51, 2, 3, 4syl3anc 1182 1  |-  ( ph  ->  ( A  x.  ( B  x.  C )
)  =  ( B  x.  ( A  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735    x. cmul 8742
This theorem is referenced by:  divrec  9440  remullem  11613  sqreulem  11843  cvgrat  12339  tanval3  12414  sinadd  12444  dvdsmulgcd  12733  prmdiv  12853  vdwlem6  13033  itgmulc2  19188  dvexp3  19325  aaliou3lem8  19725  dvradcnv  19797  pserdvlem2  19804  abelthlem6  19812  abelthlem7  19814  tangtx  19873  tanarg  19970  dvcxp1  20082  dcubic1  20141  mcubic  20143  dquart  20149  quart1  20152  quartlem1  20153  asinsin  20188  basellem3  20320  bcp1ctr  20518  lgseisenlem2  20589  lgseisenlem4  20591  lgsquadlem1  20593  2sqlem4  20606  chebbnd1lem3  20620  rpvmasum2  20661  mulog2sumlem3  20685  selberglem1  20694  selberg4lem1  20709  selberg3r  20718  selberg34r  20720  pntrlog2bndlem4  20729  pntrlog2bndlem6  20732  pntlemr  20751  pntlemk  20755  ostth2lem3  20784  branmfn  22685  colinearalglem4  24537  areacirclem2  24925  pellexlem6  26919  pell1234qrmulcl  26940  rmxyadd  27006  jm2.18  27081  jm2.19lem1  27082  jm2.22  27088  jm2.20nn  27090  proot1ex  27520  ofmul12  27542  wallispi2lem1  27820  stirlinglem1  27823  stirlinglem3  27825  stirlinglem7  27829  stirlinglem15  27837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-mulcom 8801  ax-mulass 8803
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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