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Theorem tglinerflx2 25429
Description: Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tglineelsb2.1 (𝜑𝑃𝐵)
tglineelsb2.2 (𝜑𝑄𝐵)
tglineelsb2.4 (𝜑𝑃𝑄)
Assertion
Ref Expression
tglinerflx2 (𝜑𝑄 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx2
StepHypRef Expression
1 tglineelsb2.p . 2 𝐵 = (Base‘𝐺)
2 tglineelsb2.i . 2 𝐼 = (Itv‘𝐺)
3 tglineelsb2.l . 2 𝐿 = (LineG‘𝐺)
4 tglineelsb2.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tglineelsb2.1 . 2 (𝜑𝑃𝐵)
6 tglineelsb2.2 . 2 (𝜑𝑄𝐵)
7 tglineelsb2.4 . 2 (𝜑𝑃𝑄)
8 eqid 2621 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv2 25282 . 2 (𝜑𝑄 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 6, 7, 9btwnlng1 25414 1 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wne 2790  cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-trkgc 25247  df-trkgcb 25249  df-trkg 25252
This theorem is referenced by:  tghilberti1  25432  tglnpt2  25436  colline  25444  footex  25513  foot  25514  footne  25515  perprag  25518  colperpexlem3  25524  mideulem2  25526  opphllem  25527  opphllem5  25543  opphllem6  25544  opphl  25546  outpasch  25547  hlpasch  25548  lnopp2hpgb  25555  hypcgrlem1  25591  hypcgrlem2  25592  trgcopyeulem  25597  acopy  25624  acopyeu  25625  tgasa1  25639
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