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Theorem tgcolg 25249
 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 476 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
21olcd 407 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
3 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
4 eqid 2610 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
5 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
6 tglngval.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
76adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
8 tgcolg.z . . . . . . 7 (𝜑𝑍𝑃)
98adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑍𝑃)
10 tglngval.x . . . . . . 7 (𝜑𝑋𝑃)
1110adantr 480 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋𝑃)
123, 4, 5, 7, 9, 11tgbtwntriv2 25182 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
131oveq2d 6565 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
1412, 13eleqtrd 2690 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15143mix2d 1230 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
162, 152thd 254 . 2 ((𝜑𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17 simpr 476 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝑌)
1817neneqd 2787 . . . . 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 275 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
246adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝐺 ∈ TarskiG)
2510adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝑃)
26 tglngval.y . . . . 5 (𝜑𝑌𝑃)
2726adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝑃)
288adantr 480 . . . 4 ((𝜑𝑋𝑌) → 𝑍𝑃)
293, 23, 5, 24, 25, 27, 17, 28tgellng 25248 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3022, 29bitr3d 269 . 2 ((𝜑𝑋𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3116, 30pm2.61dane 2869 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 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-trkgc 25147  df-trkgcb 25149  df-trkg 25152 This theorem is referenced by:  btwncolg1  25250  btwncolg2  25251  btwncolg3  25252  colcom  25253  colrot1  25254  lnxfr  25261  lnext  25262  tgfscgr  25263  tglowdim2l  25345  outpasch  25447
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