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Theorem tgellng 25668
Description: Property of lying on the line going through points 𝑋 and 𝑌. Definition 4.10 of [Schwabhauser] p. 36. We choose the notation 𝑍 ∈ (𝑋(LineG‘𝐺)𝑌) instead of "colinear" because LineG is a common structure slot for other axiomatizations of geometry. (Contributed by Thierry Arnoux, 28-Mar-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)
tglngval.z (𝜑𝑋𝑌)
tgellng.z (𝜑𝑍𝑃)
Assertion
Ref Expression
tgellng (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))

Proof of Theorem tgellng
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 tglngval.p . . . . 5 𝑃 = (Base‘𝐺)
2 tglngval.l . . . . 5 𝐿 = (LineG‘𝐺)
3 tglngval.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tglngval.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
5 tglngval.x . . . . 5 (𝜑𝑋𝑃)
6 tglngval.y . . . . 5 (𝜑𝑌𝑃)
7 tglngval.z . . . . 5 (𝜑𝑋𝑌)
81, 2, 3, 4, 5, 6, 7tglngval 25666 . . . 4 (𝜑 → (𝑋𝐿𝑌) = {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))})
98eleq2d 2835 . . 3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ 𝑍 ∈ {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))}))
10 eleq1 2837 . . . . 5 (𝑧 = 𝑍 → (𝑧 ∈ (𝑋𝐼𝑌) ↔ 𝑍 ∈ (𝑋𝐼𝑌)))
11 oveq1 6799 . . . . . 6 (𝑧 = 𝑍 → (𝑧𝐼𝑌) = (𝑍𝐼𝑌))
1211eleq2d 2835 . . . . 5 (𝑧 = 𝑍 → (𝑋 ∈ (𝑧𝐼𝑌) ↔ 𝑋 ∈ (𝑍𝐼𝑌)))
13 oveq2 6800 . . . . . 6 (𝑧 = 𝑍 → (𝑋𝐼𝑧) = (𝑋𝐼𝑍))
1413eleq2d 2835 . . . . 5 (𝑧 = 𝑍 → (𝑌 ∈ (𝑋𝐼𝑧) ↔ 𝑌 ∈ (𝑋𝐼𝑍)))
1510, 12, 143orbi123d 1545 . . . 4 (𝑧 = 𝑍 → ((𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧)) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
1615elrab 3513 . . 3 (𝑍 ∈ {𝑧𝑃 ∣ (𝑧 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑧𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑧))} ↔ (𝑍𝑃 ∧ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
179, 16syl6bb 276 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍𝑃 ∧ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
18 tgellng.z . . 3 (𝜑𝑍𝑃)
1918biantrurd 516 . 2 (𝜑 → ((𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)) ↔ (𝑍𝑃 ∧ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))))
2017, 19bitr4d 271 1 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3o 1069   = wceq 1630  wcel 2144  wne 2942  {crab 3064  cfv 6031  (class class class)co 6792  Basecbs 16063  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 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-trkg 25572
This theorem is referenced by:  tgcolg  25669  hlln  25722  lnhl  25730  btwnlng1  25734  btwnlng2  25735  btwnlng3  25736  lncom  25737  lnrot1  25738  lnrot2  25739  tglineeltr  25746  colmid  25803  cgracol  25939
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