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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 1339 | . 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 27804 | . 2 β’ (π β (π β (ππΏπ) β (π β (ππΌπ) β¨ π β (ππΌπ) β¨ π β (ππΌπ)))) |
12 | 2, 11 | mpbird 257 | 1 β’ (π β π β (ππΏπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ w3o 1087 = wceq 1542 β wcel 2107 β wne 2941 βcfv 6544 (class class class)co 7409 Basecbs 17144 TarskiGcstrkg 27678 Itvcitv 27684 LineGclng 27685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-trkg 27704 |
This theorem is referenced by: midexlem 27943 footexALT 27969 footexlem1 27970 footexlem2 27971 mideulem2 27985 opphllem1 27998 outpasch 28006 colhp 28021 |
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