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Theorem tglinerflx1 26413
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 2821 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv1 26271 . 2 (𝜑𝑃 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 5, 7, 9btwnlng1 26399 1 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  wne 3016  cfv 6350  (class class class)co 7150  Basecbs 16477  distcds 16568  TarskiGcstrkg 26210  Itvcitv 26216  LineGclng 26217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-trkgc 26228  df-trkgb 26229  df-trkgcb 26230  df-trkg 26233
This theorem is referenced by:  tghilberti1  26417  tglnne0  26420  tglnpt2  26421  tglineneq  26424  coltr  26427  colline  26429  footexALT  26498  footexlem1  26499  footexlem2  26500  foot  26502  footne  26503  perprag  26506  colperp  26509  colperpexlem3  26512  mideulem2  26514  outpasch  26535  hlpasch  26536  lnopp2hpgb  26543  colopp  26549  lmieu  26564  lmimid  26574  hypcgrlem1  26579  hypcgrlem2  26580  trgcopyeulem  26585  tgasa1  26638
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