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Theorem tgcolg 28577
Description: We choose the notation (𝑍 ∈ (𝑋𝐿𝑌) ∨ 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 (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tgcolg.z (𝜑𝑍𝑃)
Assertion
Ref Expression
tgcolg (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Proof of Theorem tgcolg
StepHypRef Expression
1 animorr 980 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2735 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
4 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tglngval.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 tgcolg.z . . . . . . 7 (𝜑𝑍𝑃)
87adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑍𝑃)
9 tglngval.x . . . . . . 7 (𝜑𝑋𝑃)
109adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋𝑃)
112, 3, 4, 6, 8, 10tgbtwntriv2 28510 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
12 simpr 484 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1312oveq2d 7447 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
1411, 13eleqtrd 2841 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15143mix2d 1336 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
161, 152thd 265 . 2 ((𝜑𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17 simpr 484 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝑌)
1817neneqd 2943 . . . . 5 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
19 biorf 936 . . . . 5 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
2018, 19syl 17 . . . 4 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
21 orcom 870 . . . 4 ((𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2220, 21bitrdi 287 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
245adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝐺 ∈ TarskiG)
259adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝑃)
26 tglngval.y . . . . 5 (𝜑𝑌𝑃)
2726adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝑃)
287adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑍𝑃)
292, 23, 4, 24, 25, 27, 17, 28tgellng 28576 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3022, 29bitr3d 281 . 2 ((𝜑𝑋𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3116, 30pm2.61dane 3027 1 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847  w3o 1085   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkgc 28471  df-trkgcb 28473  df-trkg 28476
This theorem is referenced by:  btwncolg1  28578  btwncolg2  28579  btwncolg3  28580  colcom  28581  colrot1  28582  lnxfr  28589  lnext  28590  tgfscgr  28591  tglowdim2l  28673  outpasch  28778
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