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Mirrors > Home > MPE Home > Th. List > tglnpt | Structured version Visualization version GIF version |
Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.) |
Ref | Expression |
---|---|
tglng.p | β’ π = (BaseβπΊ) |
tglng.l | β’ πΏ = (LineGβπΊ) |
tglng.i | β’ πΌ = (ItvβπΊ) |
tglnpt.g | β’ (π β πΊ β TarskiG) |
tglnpt.a | β’ (π β π΄ β ran πΏ) |
tglnpt.x | β’ (π β π β π΄) |
Ref | Expression |
---|---|
tglnpt | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglnpt.g | . . 3 β’ (π β πΊ β TarskiG) | |
2 | tglng.p | . . . 4 β’ π = (BaseβπΊ) | |
3 | tglng.l | . . . 4 β’ πΏ = (LineGβπΊ) | |
4 | tglng.i | . . . 4 β’ πΌ = (ItvβπΊ) | |
5 | 2, 3, 4 | tglnunirn 27830 | . . 3 β’ (πΊ β TarskiG β βͺ ran πΏ β π) |
6 | 1, 5 | syl 17 | . 2 β’ (π β βͺ ran πΏ β π) |
7 | tglnpt.a | . . . 4 β’ (π β π΄ β ran πΏ) | |
8 | elssuni 4942 | . . . 4 β’ (π΄ β ran πΏ β π΄ β βͺ ran πΏ) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β π΄ β βͺ ran πΏ) |
10 | tglnpt.x | . . 3 β’ (π β π β π΄) | |
11 | 9, 10 | sseldd 3984 | . 2 β’ (π β π β βͺ ran πΏ) |
12 | 6, 11 | sseldd 3984 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wss 3949 βͺ cuni 4909 ran crn 5678 βcfv 6544 Basecbs 17144 TarskiGcstrkg 27709 Itvcitv 27715 LineGclng 27716 |
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-cnv 5685 df-dm 5687 df-rn 5688 df-iota 6496 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-trkg 27735 |
This theorem is referenced by: mirln 27958 mirln2 27959 perpcom 27995 perpneq 27996 ragperp 27999 foot 28004 footne 28005 footeq 28006 hlperpnel 28007 perprag 28008 perpdragALT 28009 perpdrag 28010 colperpexlem3 28014 oppne3 28025 oppcom 28026 oppnid 28028 opphllem1 28029 opphllem2 28030 opphllem3 28031 opphllem4 28032 opphllem5 28033 opphllem6 28034 oppperpex 28035 opphl 28036 outpasch 28037 lnopp2hpgb 28045 hpgerlem 28047 colopp 28051 colhp 28052 lmieu 28066 lmimid 28076 lnperpex 28085 trgcopy 28086 trgcopyeulem 28087 |
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