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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 27799 | . . 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 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-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 27704 |
This theorem is referenced by: mirln 27927 mirln2 27928 perpcom 27964 perpneq 27965 ragperp 27968 foot 27973 footne 27974 footeq 27975 hlperpnel 27976 perprag 27977 perpdragALT 27978 perpdrag 27979 colperpexlem3 27983 oppne3 27994 oppcom 27995 oppnid 27997 opphllem1 27998 opphllem2 27999 opphllem3 28000 opphllem4 28001 opphllem5 28002 opphllem6 28003 oppperpex 28004 opphl 28005 outpasch 28006 lnopp2hpgb 28014 hpgerlem 28016 colopp 28020 colhp 28021 lmieu 28035 lmimid 28045 lnperpex 28054 trgcopy 28055 trgcopyeulem 28056 |
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