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Mirrors > Home > MPE Home > Th. List > btwnlng3 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
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) |
btwnlng3.1 | ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) |
Ref | Expression |
---|---|
btwnlng3 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnlng3.1 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) | |
2 | 1 | 3mix3d 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 28576 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
12 | 2, 11 | mpbird 257 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1085 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkg 28476 |
This theorem is referenced by: midexlem 28715 footexALT 28741 footexlem1 28742 footexlem2 28743 mideulem2 28757 opphllem1 28770 outpasch 28778 colhp 28793 |
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