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Theorem tglinerflx2 26428
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 2798 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv2 26281 . 2 (𝜑𝑄 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 6, 7, 9btwnlng1 26413 1 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  wne 2987  cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-trkgc 26242  df-trkgcb 26244  df-trkg 26247
This theorem is referenced by:  tghilberti1  26431  tglnpt2  26435  colline  26443  footexALT  26512  footexlem2  26514  foot  26516  footne  26517  perprag  26520  colperpexlem3  26526  mideulem2  26528  opphllem  26529  opphllem5  26545  opphllem6  26546  opphl  26548  outpasch  26549  hlpasch  26550  lnopp2hpgb  26557  hypcgrlem1  26593  hypcgrlem2  26594  trgcopyeulem  26599  acopy  26627  acopyeu  26628  tgasa1  26652
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