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Mirrors > Home > MPE Home > Th. List > tglinerflx1 | Structured version Visualization version GIF version |
Description: Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | β’ π΅ = (BaseβπΊ) |
tglineelsb2.i | β’ πΌ = (ItvβπΊ) |
tglineelsb2.l | β’ πΏ = (LineGβπΊ) |
tglineelsb2.g | β’ (π β πΊ β TarskiG) |
tglineelsb2.1 | β’ (π β π β π΅) |
tglineelsb2.2 | β’ (π β π β π΅) |
tglineelsb2.4 | β’ (π β π β π) |
Ref | Expression |
---|---|
tglinerflx1 | β’ (π β π β (ππΏπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . 2 β’ π΅ = (BaseβπΊ) | |
2 | tglineelsb2.i | . 2 β’ πΌ = (ItvβπΊ) | |
3 | tglineelsb2.l | . 2 β’ πΏ = (LineGβπΊ) | |
4 | tglineelsb2.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | tglineelsb2.1 | . 2 β’ (π β π β π΅) | |
6 | tglineelsb2.2 | . 2 β’ (π β π β π΅) | |
7 | tglineelsb2.4 | . 2 β’ (π β π β π) | |
8 | eqid 2732 | . . 3 β’ (distβπΊ) = (distβπΊ) | |
9 | 1, 8, 2, 4, 5, 6 | tgbtwntriv1 27780 | . 2 β’ (π β π β (ππΌπ)) |
10 | 1, 2, 3, 4, 5, 6, 5, 7, 9 | btwnlng1 27908 | 1 β’ (π β π β (ππΏπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β wne 2940 βcfv 6543 (class class class)co 7411 Basecbs 17146 distcds 17208 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-pw 4604 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-trkgc 27737 df-trkgb 27738 df-trkgcb 27739 df-trkg 27742 |
This theorem is referenced by: tghilberti1 27926 tglnne0 27929 tglnpt2 27930 tglineneq 27933 coltr 27936 colline 27938 footexALT 28007 footexlem1 28008 footexlem2 28009 foot 28011 footne 28012 perprag 28015 colperp 28018 colperpexlem3 28021 mideulem2 28023 outpasch 28044 hlpasch 28045 lnopp2hpgb 28052 colopp 28058 lmieu 28073 lmimid 28083 hypcgrlem1 28088 hypcgrlem2 28089 trgcopyeulem 28094 tgasa1 28147 |
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