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Mirrors > Home > MPE Home > Th. List > btwncolg2 | Structured version Visualization version GIF version |
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.) |
Ref | Expression |
---|---|
tglngval.p | β’ π = (BaseβπΊ) |
tglngval.l | β’ πΏ = (LineGβπΊ) |
tglngval.i | β’ πΌ = (ItvβπΊ) |
tglngval.g | β’ (π β πΊ β TarskiG) |
tglngval.x | β’ (π β π β π) |
tglngval.y | β’ (π β π β π) |
tgcolg.z | β’ (π β π β π) |
btwncolg2.z | β’ (π β π β (ππΌπ)) |
Ref | Expression |
---|---|
btwncolg2 | β’ (π β (π β (ππΏπ) β¨ π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwncolg2.z | . . 3 β’ (π β π β (ππΌπ)) | |
2 | 1 | 3mix2d 1334 | . 2 β’ (π β (π β (ππΌπ) β¨ π β (ππΌπ) β¨ π β (ππΌπ))) |
3 | tglngval.p | . . 3 β’ π = (BaseβπΊ) | |
4 | tglngval.l | . . 3 β’ πΏ = (LineGβπΊ) | |
5 | tglngval.i | . . 3 β’ πΌ = (ItvβπΊ) | |
6 | tglngval.g | . . 3 β’ (π β πΊ β TarskiG) | |
7 | tglngval.x | . . 3 β’ (π β π β π) | |
8 | tglngval.y | . . 3 β’ (π β π β π) | |
9 | tgcolg.z | . . 3 β’ (π β π β π) | |
10 | 3, 4, 5, 6, 7, 8, 9 | tgcolg 28402 | . 2 β’ (π β ((π β (ππΏπ) β¨ π = π) β (π β (ππΌπ) β¨ π β (ππΌπ) β¨ π β (ππΌπ)))) |
11 | 2, 10 | mpbird 256 | 1 β’ (π β (π β (ππΏπ) β¨ π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β¨ wo 845 β¨ w3o 1083 = wceq 1533 β wcel 2098 βcfv 6543 (class class class)co 7416 Basecbs 17179 TarskiGcstrkg 28275 Itvcitv 28281 LineGclng 28282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-trkgc 28296 df-trkgcb 28298 df-trkg 28301 |
This theorem is referenced by: tgdim01ln 28412 lnxfr 28414 tgbtwnconn1lem3 28422 tgbtwnconnln1 28428 tgbtwnconnln2 28429 tglineeltr 28479 |
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