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Theorem colinearperm3 30401
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 980 . . 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 11 . 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 30397 . 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 979 . . 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 30397 . . 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 473 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    \/ w3o 973    /\ w3a 974    e. wcel 1842   <.cop 3978   class class class wbr 4395   ` cfv 5569   NNcn 10576   EEcee 24608    Btwn cbtwn 24609    Colinear ccolin 30375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-iota 5533  df-fv 5577  df-oprab 6282  df-colinear 30377
This theorem is referenced by:  colinearperm2  30402  colinearperm4  30403  btwncolinear4  30410
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