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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 2731 | . . 3 β’ (distβπΊ) = (distβπΊ) | |
9 | 1, 8, 2, 4, 5, 6 | tgbtwntriv1 28006 | . 2 β’ (π β π β (ππΌπ)) |
10 | 1, 2, 3, 4, 5, 6, 5, 7, 9 | btwnlng1 28134 | 1 β’ (π β π β (ππΏπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β wne 2939 βcfv 6544 (class class class)co 7412 Basecbs 17149 distcds 17211 TarskiGcstrkg 27942 Itvcitv 27948 LineGclng 27949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 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 7415 df-oprab 7416 df-mpo 7417 df-trkgc 27963 df-trkgb 27964 df-trkgcb 27965 df-trkg 27968 |
This theorem is referenced by: tghilberti1 28152 tglnne0 28155 tglnpt2 28156 tglineneq 28159 coltr 28162 colline 28164 footexALT 28233 footexlem1 28234 footexlem2 28235 foot 28237 footne 28238 perprag 28241 colperp 28244 colperpexlem3 28247 mideulem2 28249 outpasch 28270 hlpasch 28271 lnopp2hpgb 28278 colopp 28284 lmieu 28299 lmimid 28309 hypcgrlem1 28314 hypcgrlem2 28315 trgcopyeulem 28320 tgasa1 28373 |
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