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Theorem tgcolg 25669
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 simpr 479 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
21olcd 407 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
3 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2760 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
5 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
6 tglngval.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
76adantr 472 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
8 tgcolg.z . . . . . . 7 (𝜑𝑍𝑃)
98adantr 472 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑍𝑃)
10 tglngval.x . . . . . . 7 (𝜑𝑋𝑃)
1110adantr 472 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋𝑃)
123, 4, 5, 7, 9, 11tgbtwntriv2 25602 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
131oveq2d 6830 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
1412, 13eleqtrd 2841 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15143mix2d 1422 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
162, 152thd 255 . 2 ((𝜑𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17 simpr 479 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝑌)
1817neneqd 2937 . . . . 5 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
19 biorf 419 . . . . 5 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
2018, 19syl 17 . . . 4 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
21 orcom 401 . . . 4 ((𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2220, 21syl6bb 276 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
246adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝐺 ∈ TarskiG)
2510adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝑃)
26 tglngval.y . . . . 5 (𝜑𝑌𝑃)
2726adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝑃)
288adantr 472 . . . 4 ((𝜑𝑋𝑌) → 𝑍𝑃)
293, 23, 5, 24, 25, 27, 17, 28tgellng 25668 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3022, 29bitr3d 270 . 2 ((𝜑𝑋𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3116, 30pm2.61dane 3019 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 196  wo 382  wa 383  w3o 1071   = wceq 1632  wcel 2139  wne 2932  cfv 6049  (class class class)co 6814  Basecbs 16079  distcds 16172  TarskiGcstrkg 25549  Itvcitv 25555  LineGclng 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-trkgc 25567  df-trkgcb 25569  df-trkg 25572
This theorem is referenced by:  btwncolg1  25670  btwncolg2  25671  btwncolg3  25672  colcom  25673  colrot1  25674  lnxfr  25681  lnext  25682  tgfscgr  25683  tglowdim2l  25765  outpasch  25867
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