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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 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 28561 | . 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 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkg 28461 |
| This theorem is referenced by: tglnne 28636 tglinerflx1 28641 tglinerflx2 28642 coltr3 28656 mirln2 28685 midexlem 28700 colperpexlem3 28740 mideulem2 28742 opphllem1 28755 opphllem2 28756 opphllem4 28758 hlpasch 28764 lnopp2hpgb 28771 colopp 28777 lmieu 28792 lmimid 28802 lmiisolem 28804 hypcgrlem1 28807 hypcgrlem2 28808 trgcopyeulem 28813 |
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