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Theorem mul12i 9552
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 9523 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
51, 2, 3, 4mp3an 1307 1  |-  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1362    e. wcel 1755  (class class class)co 6080   CCcc 9268    x. cmul 9275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-mulcom 9334  ax-mulass 9336
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-rex 2711  df-rab 2714  df-v 2964  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-iota 5369  df-fv 5414  df-ov 6083
This theorem is referenced by:  faclbnd4lem1  12053  decsplit  14095  root1eq1  22078  cxpeq  22080  1cubrlem  22121  efiatan2  22197  2efiatan  22198  tanatan  22199  log2ublem2  22227  log2ublem3  22228  bposlem8  22515  ax5seglem7  23004  ip1ilem  24049  ipasslem10  24062  polid2i  24382  bpoly3  28048
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