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Theorem tglinerflx2 25756
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 2769 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv2 25609 . 2 (𝜑𝑄 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 6, 7, 9btwnlng1 25741 1 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1629  wcel 2143  wne 2941  cfv 6030  (class class class)co 6791  Basecbs 16070  distcds 16164  TarskiGcstrkg 25556  Itvcitv 25562  LineGclng 25563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-trkgc 25574  df-trkgcb 25576  df-trkg 25579
This theorem is referenced by:  tghilberti1  25759  tglnpt2  25763  colline  25771  footex  25840  foot  25841  footne  25842  perprag  25845  colperpexlem3  25851  mideulem2  25853  opphllem  25854  opphllem5  25870  opphllem6  25871  opphl  25873  outpasch  25874  hlpasch  25875  lnopp2hpgb  25882  hypcgrlem1  25918  hypcgrlem2  25919  trgcopyeulem  25924  acopy  25951  acopyeu  25952  tgasa1  25966
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