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Theorem mul12i 9667
Description: Commutative/associative law that swaps the first two factors in a triple product. (Contributed by NM, 11-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Hypotheses
Ref Expression
mul.1  |-  A  e.  CC
mul.2  |-  B  e.  CC
mul.3  |-  C  e.  CC
Assertion
Ref Expression
mul12i  |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
)

Proof of Theorem mul12i
StepHypRef Expression
1 mul.1 . 2  |-  A  e.  CC
2 mul.2 . 2  |-  B  e.  CC
3 mul.3 . 2  |-  C  e.  CC
4 mul12 9638 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
51, 2, 3, 4mp3an 1315 1  |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758  (class class class)co 6192   CCcc 9383    x. cmul 9390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-mulcom 9449  ax-mulass 9451
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by:  faclbnd4lem1  12172  decsplit  14216  root1eq1  22311  cxpeq  22313  1cubrlem  22354  efiatan2  22430  2efiatan  22431  tanatan  22432  log2ublem2  22460  log2ublem3  22461  bposlem8  22748  ax5seglem7  23318  ip1ilem  24363  ipasslem10  24376  polid2i  24696  bpoly3  28337
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