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| Mirrors > Home > MPE Home > Th. List > btwnlng2 | 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) |
| btwnlng2.1 | ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) |
| Ref | Expression |
|---|---|
| btwnlng2 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | btwnlng2.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌)) | |
| 2 | 1 | 3mix2d 1338 | . 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: mirln 28684 colperpexlem3 28740 outpasch 28763 hpgerlem 28773 |
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