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Theorem tglnpt 26263
Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.)
Hypotheses
Ref Expression
tglng.p 𝑃 = (Base‘𝐺)
tglng.l 𝐿 = (LineG‘𝐺)
tglng.i 𝐼 = (Itv‘𝐺)
tglnpt.g (𝜑𝐺 ∈ TarskiG)
tglnpt.a (𝜑𝐴 ∈ ran 𝐿)
tglnpt.x (𝜑𝑋𝐴)
Assertion
Ref Expression
tglnpt (𝜑𝑋𝑃)

Proof of Theorem tglnpt
StepHypRef Expression
1 tglnpt.g . . 3 (𝜑𝐺 ∈ TarskiG)
2 tglng.p . . . 4 𝑃 = (Base‘𝐺)
3 tglng.l . . . 4 𝐿 = (LineG‘𝐺)
4 tglng.i . . . 4 𝐼 = (Itv‘𝐺)
52, 3, 4tglnunirn 26262 . . 3 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
61, 5syl 17 . 2 (𝜑 ran 𝐿𝑃)
7 tglnpt.a . . . 4 (𝜑𝐴 ∈ ran 𝐿)
8 elssuni 4861 . . . 4 (𝐴 ∈ ran 𝐿𝐴 ran 𝐿)
97, 8syl 17 . . 3 (𝜑𝐴 ran 𝐿)
10 tglnpt.x . . 3 (𝜑𝑋𝐴)
119, 10sseldd 3967 . 2 (𝜑𝑋 ran 𝐿)
126, 11sseldd 3967 1 (𝜑𝑋𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wss 3935   cuni 4832  ran crn 5550  cfv 6349  Basecbs 16473  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-iota 6308  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-trkg 26167
This theorem is referenced by:  mirln  26390  mirln2  26391  perpcom  26427  perpneq  26428  ragperp  26431  foot  26436  footne  26437  footeq  26438  hlperpnel  26439  perprag  26440  perpdragALT  26441  perpdrag  26442  colperpexlem3  26446  oppne3  26457  oppcom  26458  oppnid  26460  opphllem1  26461  opphllem2  26462  opphllem3  26463  opphllem4  26464  opphllem5  26465  opphllem6  26466  oppperpex  26467  opphl  26468  outpasch  26469  lnopp2hpgb  26477  hpgerlem  26479  colopp  26483  colhp  26484  lmieu  26498  lmimid  26508  lnperpex  26517  trgcopy  26518  trgcopyeulem  26519
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