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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 1333 | . 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 28477 | . 2 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))) |
12 | 2, 11 | mpbird 256 | 1 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1083 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 TarskiGcstrkg 28351 Itvcitv 28357 LineGclng 28358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-trkg 28377 |
This theorem is referenced by: tglnne 28552 tglinerflx1 28557 tglinerflx2 28558 coltr3 28572 mirln2 28601 midexlem 28616 colperpexlem3 28656 mideulem2 28658 opphllem1 28671 opphllem2 28672 opphllem4 28674 hlpasch 28680 lnopp2hpgb 28687 colopp 28693 lmieu 28708 lmimid 28718 lmiisolem 28720 hypcgrlem1 28723 hypcgrlem2 28724 trgcopyeulem 28729 |
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