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Mirrors > Home > MPE Home > Th. List > btwnlng3 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
Ref | Expression |
---|---|
btwnlng1.p | β’ π = (BaseβπΊ) |
btwnlng1.i | β’ πΌ = (ItvβπΊ) |
btwnlng1.l | β’ πΏ = (LineGβπΊ) |
btwnlng1.g | β’ (π β πΊ β TarskiG) |
btwnlng1.x | β’ (π β π β π) |
btwnlng1.y | β’ (π β π β π) |
btwnlng1.z | β’ (π β π β π) |
btwnlng1.d | β’ (π β π β π) |
btwnlng3.1 | β’ (π β π β (ππΌπ)) |
Ref | Expression |
---|---|
btwnlng3 | β’ (π β π β (ππΏπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnlng3.1 | . . 3 β’ (π β π β (ππΌπ)) | |
2 | 1 | 3mix3d 1338 | . 2 β’ (π β (π β (ππΌπ) β¨ π β (ππΌπ) β¨ π β (ππΌπ))) |
3 | btwnlng1.p | . . 3 β’ π = (BaseβπΊ) | |
4 | btwnlng1.l | . . 3 β’ πΏ = (LineGβπΊ) | |
5 | btwnlng1.i | . . 3 β’ πΌ = (ItvβπΊ) | |
6 | btwnlng1.g | . . 3 β’ (π β πΊ β TarskiG) | |
7 | btwnlng1.x | . . 3 β’ (π β π β π) | |
8 | btwnlng1.y | . . 3 β’ (π β π β π) | |
9 | btwnlng1.d | . . 3 β’ (π β π β π) | |
10 | btwnlng1.z | . . 3 β’ (π β π β π) | |
11 | 3, 4, 5, 6, 7, 8, 9, 10 | tgellng 27842 | . 2 β’ (π β (π β (ππΏπ) β (π β (ππΌπ) β¨ π β (ππΌπ) β¨ π β (ππΌπ)))) |
12 | 2, 11 | mpbird 256 | 1 β’ (π β π β (ππΏπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1086 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6543 (class class class)co 7411 Basecbs 17146 TarskiGcstrkg 27716 Itvcitv 27722 LineGclng 27723 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-trkg 27742 |
This theorem is referenced by: midexlem 27981 footexALT 28007 footexlem1 28008 footexlem2 28009 mideulem2 28023 opphllem1 28036 outpasch 28044 colhp 28059 |
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