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Theorem btwncolg1 23663
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 1166 . 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 23662 . 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 232 1  |-  ( ph  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    \/ w3o 967    = wceq 1374    e. wcel 1762   ` cfv 5579  (class class class)co 6275   Basecbs 14479  TarskiGcstrkg 23546  Itvcitv 23553  LineGclng 23554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-trkgc 23565  df-trkgcb 23567  df-trkg 23571
This theorem is referenced by:  tgdim01ln  23672  lnxfr  23673  tgbtwnconn1lem3  23681  tgbtwnconnln3  23685  legov2  23693  tglineeltr  23718  symquadlem  23767  midexlem  23770  ragflat  23782  colperpexlem1  23802  mideulem  23806  hypcgrlem1  23834
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