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Theorem btwncolg1 24460
Description: Betweenness implies colinearity (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglngval.p  |-  P  =  ( Base `  G
)
tglngval.l  |-  L  =  (LineG `  G )
tglngval.i  |-  I  =  (Itv `  G )
tglngval.g  |-  ( ph  ->  G  e. TarskiG )
tglngval.x  |-  ( ph  ->  X  e.  P )
tglngval.y  |-  ( ph  ->  Y  e.  P )
tgcolg.z  |-  ( ph  ->  Z  e.  P )
btwncolg1.z  |-  ( ph  ->  Z  e.  ( X I Y ) )
Assertion
Ref Expression
btwncolg1  |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )

Proof of Theorem btwncolg1
StepHypRef Expression
1 btwncolg1.z . . 3  |-  ( ph  ->  Z  e.  ( X I Y ) )
213mix1d 1180 . 2  |-  ( ph  ->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) )
3 tglngval.p . . 3  |-  P  =  ( Base `  G
)
4 tglngval.l . . 3  |-  L  =  (LineG `  G )
5 tglngval.i . . 3  |-  I  =  (Itv `  G )
6 tglngval.g . . 3  |-  ( ph  ->  G  e. TarskiG )
7 tglngval.x . . 3  |-  ( ph  ->  X  e.  P )
8 tglngval.y . . 3  |-  ( ph  ->  Y  e.  P )
9 tgcolg.z . . 3  |-  ( ph  ->  Z  e.  P )
103, 4, 5, 6, 7, 8, 9tgcolg 24459 . 2  |-  ( ph  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
112, 10mpbird 235 1  |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    \/ w3o 981    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   Basecbs 15084  TarskiGcstrkg 24341  Itvcitv 24347  LineGclng 24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-trkgc 24359  df-trkgcb 24361  df-trkg 24364
This theorem is referenced by:  tgdim01ln  24469  lnxfr  24471  tgbtwnconn1lem3  24479  tgbtwnconnln3  24483  legov2  24491  ncolne1  24529  tglineeltr  24535  symquadlem  24591  midexlem  24594  ragflat  24606  colperpexlem1  24629  opphllem  24634
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