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Mirrors > Home > MPE Home > Th. List > btwnlng1 | 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) |
btwnlng1.1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) |
Ref | Expression |
---|---|
btwnlng1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnlng1.1 | . . 3 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐼𝑌)) | |
2 | 1 | 3mix1d 1328 | . 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 26267 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
12 | 2, 11 | mpbird 258 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 TarskiGcstrkg 26144 Itvcitv 26150 LineGclng 26151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-trkg 26167 |
This theorem is referenced by: tglnne 26342 tglinerflx1 26347 tglinerflx2 26348 coltr3 26362 mirln2 26391 midexlem 26406 colperpexlem3 26446 mideulem2 26448 opphllem1 26461 opphllem2 26462 opphllem4 26464 hlpasch 26470 lnopp2hpgb 26477 colopp 26483 lmieu 26498 lmimid 26508 lmiisolem 26510 hypcgrlem1 26513 hypcgrlem2 26514 trgcopyeulem 26519 |
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