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Theorem tgcolg 26351
 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 976 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2 tglngval.p . . . . . 6 𝑃 = (Base‘𝐺)
3 eqid 2801 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
4 tglngval.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tglngval.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65adantr 484 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝐺 ∈ TarskiG)
7 tgcolg.z . . . . . . 7 (𝜑𝑍𝑃)
87adantr 484 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑍𝑃)
9 tglngval.x . . . . . . 7 (𝜑𝑋𝑃)
109adantr 484 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋𝑃)
112, 3, 4, 6, 8, 10tgbtwntriv2 26284 . . . . 5 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑋))
12 simpr 488 . . . . . 6 ((𝜑𝑋 = 𝑌) → 𝑋 = 𝑌)
1312oveq2d 7155 . . . . 5 ((𝜑𝑋 = 𝑌) → (𝑍𝐼𝑋) = (𝑍𝐼𝑌))
1411, 13eleqtrd 2895 . . . 4 ((𝜑𝑋 = 𝑌) → 𝑋 ∈ (𝑍𝐼𝑌))
15143mix2d 1334 . . 3 ((𝜑𝑋 = 𝑌) → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
161, 152thd 268 . 2 ((𝜑𝑋 = 𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
17 simpr 488 . . . . . 6 ((𝜑𝑋𝑌) → 𝑋𝑌)
1817neneqd 2995 . . . . 5 ((𝜑𝑋𝑌) → ¬ 𝑋 = 𝑌)
19 biorf 934 . . . . 5 𝑋 = 𝑌 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
2018, 19syl 17 . . . 4 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌))))
21 orcom 867 . . . 4 ((𝑋 = 𝑌𝑍 ∈ (𝑋𝐿𝑌)) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
2220, 21syl6bb 290 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)))
23 tglngval.l . . . 4 𝐿 = (LineG‘𝐺)
245adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝐺 ∈ TarskiG)
259adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑋𝑃)
26 tglngval.y . . . . 5 (𝜑𝑌𝑃)
2726adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑌𝑃)
287adantr 484 . . . 4 ((𝜑𝑋𝑌) → 𝑍𝑃)
292, 23, 4, 24, 25, 27, 17, 28tgellng 26350 . . 3 ((𝜑𝑋𝑌) → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3022, 29bitr3d 284 . 2 ((𝜑𝑋𝑌) → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
3116, 30pm2.61dane 3077 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 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  distcds 16569  TarskiGcstrkg 26227  Itvcitv 26233  LineGclng 26234 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-trkgc 26245  df-trkgcb 26247  df-trkg 26250 This theorem is referenced by:  btwncolg1  26352  btwncolg2  26353  btwncolg3  26354  colcom  26355  colrot1  26356  lnxfr  26363  lnext  26364  tgfscgr  26365  tglowdim2l  26447  outpasch  26552
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