Theorem btwnlng1 28598

Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)

Hypotheses

Ref	Expression
btwnlng1.p	⊢ 𝑃 = (Base‘𝐺)
btwnlng1.i	⊢ 𝐼 = (Itv‘𝐺)
btwnlng1.l	⊢ 𝐿 = (LineG‘𝐺)
btwnlng1.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
btwnlng1.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
btwnlng1.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
btwnlng1.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
btwnlng1.d	⊢ (𝜑 → 𝑋 ≠ 𝑌)
btwnlng1.1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌))

Assertion

Ref	Expression
btwnlng1	⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng1

Step	Hyp	Ref	Expression
1		btwnlng1.1	. . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌))
2	1	3mix1d 1337	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
3		btwnlng1.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
4		btwnlng1.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		btwnlng1.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
6		btwnlng1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		btwnlng1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
8		btwnlng1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
9		btwnlng1.d	. . 3 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
10		btwnlng1.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
11	3, 4, 5, 6, 7, 8, 9, 10	tgellng 28532	. 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
12	2, 11	mpbird 257	1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌))

Syntax hints:   → wi 4   ∨ w3o 1085   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  TarskiGcstrkg 28406  Itvcitv 28412  LineGclng 28413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-trkg 28432

This theorem is referenced by:  tglnne  28607  tglinerflx1  28612  tglinerflx2  28613  coltr3  28627  mirln2  28656  midexlem  28671  colperpexlem3  28711  mideulem2  28713  opphllem1  28726  opphllem2  28727  opphllem4  28729  hlpasch  28735  lnopp2hpgb  28742  colopp  28748  lmieu  28763  lmimid  28773  lmiisolem  28775  hypcgrlem1  28778  hypcgrlem2  28779  trgcopyeulem  28784