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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 1335 | . 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 26348 | . 2 ⊢ (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
11 | 2, 10 | mpbird 260 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 844 ∨ w3o 1083 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: tgdim01ln 26358 lnxfr 26360 tgidinside 26365 tgbtwnconn1lem3 26368 tgbtwnconnln3 26372 tgbtwnconnln1 26374 tgbtwnconnln2 26375 legov 26379 legov2 26380 legtrd 26383 tglineeltr 26425 krippenlem 26484 midexlem 26486 footexALT 26512 footexlem2 26514 mideulem2 26528 hlpasch 26550 hypcgrlem1 26593 cgracol 26622 |
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