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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 1231 | . 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 25249 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
11 | 2, 10 | mpbird 246 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-trkgc 25147 df-trkgcb 25149 df-trkg 25152 |
This theorem is referenced by: tgdim01ln 25259 lnxfr 25261 tgidinside 25266 tgbtwnconn1lem3 25269 tgbtwnconnln3 25273 tgbtwnconnln1 25275 tgbtwnconnln2 25276 legov 25280 legov2 25281 legtrd 25284 tglineeltr 25326 krippenlem 25385 midexlem 25387 footex 25413 mideulem2 25426 hlpasch 25448 hypcgrlem1 25491 cgracol 25519 |
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