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Theorem tgcolg 24324
Description: We choose the notation  ( Z  e.  ( X L Y )  \/  X  =  Y ) instead of "colinear" in order to avoid defining an additional symbol for colinearity because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 25-May-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 )
Assertion
Ref Expression
tgcolg  |-  ( ph  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )

Proof of Theorem tgcolg
StepHypRef Expression
1 simpr 459 . . . 4  |-  ( (
ph  /\  X  =  Y )  ->  X  =  Y )
21olcd 391 . . 3  |-  ( (
ph  /\  X  =  Y )  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
3 tglngval.p . . . . . 6  |-  P  =  ( Base `  G
)
4 eqid 2402 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
5 tglngval.i . . . . . 6  |-  I  =  (Itv `  G )
6 tglngval.g . . . . . . 7  |-  ( ph  ->  G  e. TarskiG )
76adantr 463 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  G  e. TarskiG )
8 tgcolg.z . . . . . . 7  |-  ( ph  ->  Z  e.  P )
98adantr 463 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  Z  e.  P )
10 tglngval.x . . . . . . 7  |-  ( ph  ->  X  e.  P )
1110adantr 463 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  P )
123, 4, 5, 7, 9, 11tgbtwntriv2 24259 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I X ) )
131oveq2d 6294 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  ( Z I X )  =  ( Z I Y ) )
1412, 13eleqtrd 2492 . . . 4  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I Y ) )
15143mix2d 1173 . . 3  |-  ( (
ph  /\  X  =  Y )  ->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) )
162, 152thd 240 . 2  |-  ( (
ph  /\  X  =  Y )  ->  (
( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
17 simpr 459 . . . . . 6  |-  ( (
ph  /\  X  =/=  Y )  ->  X  =/=  Y )
1817neneqd 2605 . . . . 5  |-  ( (
ph  /\  X  =/=  Y )  ->  -.  X  =  Y )
19 biorf 403 . . . . 5  |-  ( -.  X  =  Y  -> 
( Z  e.  ( X L Y )  <-> 
( X  =  Y  \/  Z  e.  ( X L Y ) ) ) )
2018, 19syl 17 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  ( Z  e.  ( X L Y )  <->  ( X  =  Y  \/  Z  e.  ( X L Y ) ) ) )
21 orcom 385 . . . 4  |-  ( ( X  =  Y  \/  Z  e.  ( X L Y ) )  <->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
2220, 21syl6bb 261 . . 3  |-  ( (
ph  /\  X  =/=  Y )  ->  ( Z  e.  ( X L Y )  <->  ( Z  e.  ( X L Y )  \/  X  =  Y ) ) )
23 tglngval.l . . . 4  |-  L  =  (LineG `  G )
246adantr 463 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  G  e. TarskiG )
2510adantr 463 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  X  e.  P )
26 tglngval.y . . . . 5  |-  ( ph  ->  Y  e.  P )
2726adantr 463 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  Y  e.  P )
288adantr 463 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  Z  e.  P )
293, 23, 5, 24, 25, 27, 17, 28tgellng 24323 . . 3  |-  ( (
ph  /\  X  =/=  Y )  ->  ( Z  e.  ( X L Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
3022, 29bitr3d 255 . 2  |-  ( (
ph  /\  X  =/=  Y )  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
3116, 30pm2.61dane 2721 1  |-  ( ph  ->  ( ( Z  e.  ( X L Y )  \/  X  =  Y )  <->  ( Z  e.  ( X I Y )  \/  X  e.  ( Z I Y )  \/  Y  e.  ( X I Z ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5569  (class class class)co 6278   Basecbs 14841   distcds 14918  TarskiGcstrkg 24206  Itvcitv 24212  LineGclng 24213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-trkgc 24224  df-trkgcb 24226  df-trkg 24229
This theorem is referenced by:  btwncolg1  24325  btwncolg2  24326  btwncolg3  24327  colcom  24328  colrot1  24329  lnxfr  24336  lnext  24337  tgfscgr  24338  tglowdim2l  24416
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