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Theorem tglinerflx1 26346
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 2818 . . 3 (dist‘𝐺) = (dist‘𝐺)
91, 8, 2, 4, 5, 6tgbtwntriv1 26204 . 2 (𝜑𝑃 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 5, 7, 9btwnlng1 26332 1 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wne 3013  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149  LineGclng 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  tghilberti1  26350  tglnne0  26353  tglnpt2  26354  tglineneq  26357  coltr  26360  colline  26362  footexALT  26431  footexlem1  26432  footexlem2  26433  foot  26435  footne  26436  perprag  26439  colperp  26442  colperpexlem3  26445  mideulem2  26447  outpasch  26468  hlpasch  26469  lnopp2hpgb  26476  colopp  26482  lmieu  26497  lmimid  26507  hypcgrlem1  26512  hypcgrlem2  26513  trgcopyeulem  26518  tgasa1  26571
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