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Theorem colinearperm3 24093
Description: Permutation law for colinearity. Part of theorem 4.11 of [Schwabhauser] p. 36. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
colinearperm3  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  B  Colinear  <. C ,  A >. ) )

Proof of Theorem colinearperm3
StepHypRef Expression
1 3orrot 945 . . 3  |-  ( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  <->  ( B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  C >. ) )
21a1i 12 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. )  <-> 
( B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  C >. ) ) )
3 brcolinear 24089 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  ( A  Btwn  <. B ,  C >.  \/  B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >. ) ) )
4 3anrot 944 . . 3  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) )  <->  ( B  e.  ( EE `  N
)  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N ) ) )
5 brcolinear 24089 . . 3  |-  ( ( N  e.  NN  /\  ( B  e.  ( EE `  N )  /\  C  e.  ( EE `  N )  /\  A  e.  ( EE `  N
) ) )  -> 
( B  Colinear  <. C ,  A >. 
<->  ( B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  C >. ) ) )
64, 5sylan2b 463 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( B  Colinear  <. C ,  A >. 
<->  ( B  Btwn  <. C ,  A >.  \/  C  Btwn  <. A ,  B >.  \/  A  Btwn  <. B ,  C >. ) ) )
72, 3, 63bitr4d 278 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  B  Colinear  <. C ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    \/ w3o 938    /\ w3a 939    e. wcel 1688   <.cop 3644   class class class wbr 4024   ` cfv 5221   NNcn 9741   EEcee 23923    Btwn cbtwn 23924    Colinear ccolin 24067
This theorem is referenced by:  colinearperm2  24094  colinearperm4  24095  btwncolinear4  24102
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-oprab 5823  df-colinear 24071
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