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Theorem btwncolinear4 25713
Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
btwncolinear4  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  B  Colinear  <. C ,  A >. ) )

Proof of Theorem btwncolinear4
StepHypRef Expression
1 btwncolinear1 25710 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  A  Colinear  <. B ,  C >. ) )
2 colinearperm3 25704 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( A  Colinear  <. B ,  C >. 
<->  B  Colinear  <. C ,  A >. ) )
31, 2sylibd 206 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  C  e.  ( EE `  N
) ) )  -> 
( C  Btwn  <. A ,  B >.  ->  B  Colinear  <. C ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1717   <.cop 3753   class class class wbr 4146   ` cfv 5387   NNcn 9925   EEcee 25534    Btwn cbtwn 25535    Colinear ccolin 25678
This theorem is referenced by:  btwnconn1lem12  25739  lineunray  25788  lineelsb2  25789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-iota 5351  df-fv 5395  df-oprab 6017  df-colinear 25682
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