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| Mirrors > Home > MPE Home > Th. List > btwncolg3 | Structured version Visualization version GIF version | ||
| Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
| Ref | Expression |
|---|---|
| tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
| tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
| tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| tgcolg.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
| btwncolg3.z | ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) |
| Ref | Expression |
|---|---|
| btwncolg3 | ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | btwncolg3.z | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍)) | |
| 2 | 1 | 3mix3d 1339 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))) |
| 3 | tglngval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 4 | tglngval.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
| 5 | tglngval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 6 | tglngval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 7 | tglngval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
| 8 | tglngval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 9 | tgcolg.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃) | |
| 10 | 3, 4, 5, 6, 7, 8, 9 | tgcolg 28562 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
| 11 | 2, 10 | mpbird 257 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ‘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-trkgc 28456 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: tgdim01ln 28572 lnxfr 28574 tgidinside 28579 tgbtwnconn1lem3 28582 tgbtwnconnln3 28586 tgbtwnconnln1 28588 tgbtwnconnln2 28589 legov 28593 legov2 28594 legtrd 28597 tglineeltr 28639 krippenlem 28698 midexlem 28700 footexALT 28726 footexlem2 28728 mideulem2 28742 hlpasch 28764 hypcgrlem1 28807 cgracol 28836 |
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