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Theorem mul12d 9037
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 8994 . 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 1632    e. wcel 1696  (class class class)co 5874   CCcc 8751    x. cmul 8758
This theorem is referenced by:  divrec  9456  remullem  11629  sqreulem  11859  cvgrat  12355  tanval3  12430  sinadd  12460  dvdsmulgcd  12749  prmdiv  12869  vdwlem6  13049  itgmulc2  19204  dvexp3  19341  aaliou3lem8  19741  dvradcnv  19813  pserdvlem2  19820  abelthlem6  19828  abelthlem7  19830  tangtx  19889  tanarg  19986  dvcxp1  20098  dcubic1  20157  mcubic  20159  dquart  20165  quart1  20168  quartlem1  20169  asinsin  20204  basellem3  20336  bcp1ctr  20534  lgseisenlem2  20605  lgseisenlem4  20607  lgsquadlem1  20609  2sqlem4  20622  chebbnd1lem3  20636  rpvmasum2  20677  mulog2sumlem3  20701  selberglem1  20710  selberg4lem1  20725  selberg3r  20734  selberg34r  20736  pntrlog2bndlem4  20745  pntrlog2bndlem6  20748  pntlemr  20767  pntlemk  20771  ostth2lem3  20800  branmfn  22701  faclimlem9  24125  colinearalglem4  24609  itgmulc2nc  25019  areacirclem2  25028  pellexlem6  27022  pell1234qrmulcl  27043  rmxyadd  27109  jm2.18  27184  jm2.19lem1  27185  jm2.22  27191  jm2.20nn  27193  proot1ex  27623  ofmul12  27645  wallispi2lem1  27923  stirlinglem1  27926  stirlinglem3  27928  stirlinglem7  27932  stirlinglem15  27940
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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