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Theorem tglinerflx1 25423
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
tglinerflx1 (𝜑𝑃 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx1
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 2626 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv1 25281 . 2 (𝜑𝑃 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 5, 7, 9btwnlng1 25409 1 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  wne 2796  cfv 5850  (class class class)co 6605  Basecbs 15776  distcds 15866  TarskiGcstrkg 25224  Itvcitv 25230  LineGclng 25231
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-trkgc 25242  df-trkgb 25243  df-trkgcb 25244  df-trkg 25247
This theorem is referenced by:  tghilberti1  25427  tglnne0  25430  tglnpt2  25431  tglineneq  25434  coltr  25437  colline  25439  footex  25508  foot  25509  footne  25510  perprag  25513  colperp  25516  colperpexlem3  25519  mideulem2  25521  outpasch  25542  hlpasch  25543  lnopp2hpgb  25550  colopp  25556  lmieu  25571  lmimid  25581  hypcgrlem1  25586  hypcgrlem2  25587  trgcopyeulem  25592  tgasa1  25634
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