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Theorem btwnlng1 24203
Description: Betweenness implies colinearity (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
btwnlng1.p  |-  P  =  ( Base `  G
)
btwnlng1.i  |-  I  =  (Itv `  G )
btwnlng1.l  |-  L  =  (LineG `  G )
btwnlng1.g  |-  ( ph  ->  G  e. TarskiG )
btwnlng1.x  |-  ( ph  ->  X  e.  P )
btwnlng1.y  |-  ( ph  ->  Y  e.  P )
btwnlng1.z  |-  ( ph  ->  Z  e.  P )
btwnlng1.d  |-  ( ph  ->  X  =/=  Y )
btwnlng1.1  |-  ( ph  ->  Z  e.  ( X I Y ) )
Assertion
Ref Expression
btwnlng1  |-  ( ph  ->  Z  e.  ( X L Y ) )

Proof of Theorem btwnlng1
StepHypRef Expression
1 btwnlng1.1 . . 3  |-  ( ph  ->  Z  e.  ( X I Y ) )
213mix1d 1169 . 2  |-  ( ph  ->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) )
3 btwnlng1.p . . 3  |-  P  =  ( Base `  G
)
4 btwnlng1.l . . 3  |-  L  =  (LineG `  G )
5 btwnlng1.i . . 3  |-  I  =  (Itv `  G )
6 btwnlng1.g . . 3  |-  ( ph  ->  G  e. TarskiG )
7 btwnlng1.x . . 3  |-  ( ph  ->  X  e.  P )
8 btwnlng1.y . . 3  |-  ( ph  ->  Y  e.  P )
9 btwnlng1.d . . 3  |-  ( ph  ->  X  =/=  Y )
10 btwnlng1.z . . 3  |-  ( ph  ->  Z  e.  P )
113, 4, 5, 6, 7, 8, 9, 10tgellng 24144 . 2  |-  ( ph  ->  ( Z  e.  ( X L Y )  <-> 
( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
122, 11mpbird 232 1  |-  ( ph  ->  Z  e.  ( X L Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 970    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   Basecbs 14719  TarskiGcstrkg 24026  Itvcitv 24033  LineGclng 24034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-trkg 24051
This theorem is referenced by:  tglnne  24212  tglinerflx1  24217  tglinerflx2  24218  coltr3  24233  mirln2  24261  midexlem  24273  colperpexlem3  24310  mideulem2  24312  opphllem1  24323  opphllem2  24324  opphllem4  24326  lnopp2hpgb  24336  lmieu  24354  lmimid  24363  lmiisolem  24365  hypcgrlem1  24368  hypcgrlem2  24369
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