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Theorem tgcolg 23664
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 an 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 461 . . . 4  |-  ( (
ph  /\  X  =  Y )  ->  X  =  Y )
21olcd 393 . . 3  |-  ( (
ph  /\  X  =  Y )  ->  ( Z  e.  ( X L Y )  \/  X  =  Y ) )
3 tglngval.p . . . . . 6  |-  P  =  ( Base `  G
)
4 eqid 2462 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
5 tglngval.i . . . . . 6  |-  I  =  (Itv `  G )
6 tglngval.g . . . . . . 7  |-  ( ph  ->  G  e. TarskiG )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  G  e. TarskiG )
8 tgcolg.z . . . . . . 7  |-  ( ph  ->  Z  e.  P )
98adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  Z  e.  P )
10 tglngval.x . . . . . . 7  |-  ( ph  ->  X  e.  P )
1110adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  P )
123, 4, 5, 7, 9, 11tgbtwntriv2 23601 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I X ) )
131oveq2d 6293 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  ( Z I X )  =  ( Z I Y ) )
1412, 13eleqtrd 2552 . . . 4  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I Y ) )
15143mix2d 1167 . . 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 461 . . . . . 6  |-  ( (
ph  /\  X  =/=  Y )  ->  X  =/=  Y )
1817neneqd 2664 . . . . 5  |-  ( (
ph  /\  X  =/=  Y )  ->  -.  X  =  Y )
19 biorf 405 . . . . 5  |-  ( -.  X  =  Y  -> 
( Z  e.  ( X L Y )  <-> 
( X  =  Y  \/  Z  e.  ( X L Y ) ) ) )
2018, 19syl 16 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  ( Z  e.  ( X L Y )  <->  ( X  =  Y  \/  Z  e.  ( X L Y ) ) ) )
21 orcom 387 . . . 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 465 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  G  e. TarskiG )
2510adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  X  e.  P )
26 tglngval.y . . . . 5  |-  ( ph  ->  Y  e.  P )
2726adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  Y  e.  P )
288adantr 465 . . . 4  |-  ( (
ph  /\  X  =/=  Y )  ->  Z  e.  P )
293, 23, 5, 24, 25, 27, 17, 28tgellng 23663 . . 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 2780 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 368    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762    =/= wne 2657   ` cfv 5581  (class class class)co 6277   Basecbs 14481   distcds 14555  TarskiGcstrkg 23548  Itvcitv 23555  LineGclng 23556
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-trkgc 23567  df-trkgcb 23569  df-trkg 23573
This theorem is referenced by:  btwncolg1  23665  btwncolg2  23666  btwncolg3  23667  colcom  23668  colrot1  23669  lnxfr  23675  lnext  23676  tgfscgr  23677  tglowdim2l  23739
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