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Theorem tglnpt 25664
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 25663 . . 3 (𝐺 ∈ TarskiG → ran 𝐿𝑃)
61, 5syl 17 . 2 (𝜑 ran 𝐿𝑃)
7 tglnpt.a . . . 4 (𝜑𝐴 ∈ ran 𝐿)
8 elssuni 4619 . . . 4 (𝐴 ∈ ran 𝐿𝐴 ran 𝐿)
97, 8syl 17 . . 3 (𝜑𝐴 ran 𝐿)
10 tglnpt.x . . 3 (𝜑𝑋𝐴)
119, 10sseldd 3745 . 2 (𝜑𝑋 ran 𝐿)
126, 11sseldd 3745 1 (𝜑𝑋𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wss 3715   cuni 4588  ran crn 5267  cfv 6049  Basecbs 16079  TarskiGcstrkg 25549  Itvcitv 25555  LineGclng 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-cnv 5274  df-dm 5276  df-rn 5277  df-iota 6012  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-trkg 25572
This theorem is referenced by:  mirln  25791  mirln2  25792  perpcom  25828  perpneq  25829  ragperp  25832  foot  25834  footne  25835  footeq  25836  hlperpnel  25837  perprag  25838  perpdragALT  25839  perpdrag  25840  colperpexlem3  25844  oppne3  25855  oppcom  25856  oppnid  25858  opphllem1  25859  opphllem2  25860  opphllem3  25861  opphllem4  25862  opphllem5  25863  opphllem6  25864  oppperpex  25865  opphl  25866  outpasch  25867  lnopp2hpgb  25875  hpgerlem  25877  colopp  25881  colhp  25882  lmieu  25896  lmimid  25906  lnperpex  25915  trgcopy  25916  trgcopyeulem  25917
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