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Theorem tgcolg 22987
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 2442 . . . . . 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 22940 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I X ) )
131oveq2d 6106 . . . . 5  |-  ( (
ph  /\  X  =  Y )  ->  ( Z I X )  =  ( Z I Y ) )
1412, 13eleqtrd 2518 . . . 4  |-  ( (
ph  /\  X  =  Y )  ->  X  e.  ( Z I Y ) )
15143mix2d 1164 . . 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 2623 . . . . 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 22986 . . 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 2688 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 964    = wceq 1369    e. wcel 1756    =/= wne 2605   ` cfv 5417  (class class class)co 6090   Basecbs 14173   distcds 14246  TarskiGcstrkg 22888  Itvcitv 22896  LineGclng 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-trkgc 22908  df-trkgcb 22910  df-trkg 22915
This theorem is referenced by:  btwncolg1  22988  btwncolg2  22989  btwncolg3  22990  colcom  22991  colrot1  22992  lnxfr  22997  lnext  22998  tgfscgr  22999  tglowdim2l  23052
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