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Theorem colinearperm3 25945
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 942 . . 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 25941 . 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 941 . . 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 25941 . . 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 462 . 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 277 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 4    <-> wb 177    /\ wa 359    \/ w3o 935    /\ w3a 936    e. wcel 1725   <.cop 3809   class class class wbr 4204   ` cfv 5445   NNcn 9989   EEcee 25775    Btwn cbtwn 25776    Colinear ccolin 25919
This theorem is referenced by:  colinearperm2  25946  colinearperm4  25947  btwncolinear4  25954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-iota 5409  df-fv 5453  df-oprab 6076  df-colinear 25923
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