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Theorem tglinerflx1 27884
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 𝐡 = (Baseβ€˜πΊ)
tglineelsb2.i 𝐼 = (Itvβ€˜πΊ)
tglineelsb2.l 𝐿 = (LineGβ€˜πΊ)
tglineelsb2.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglineelsb2.1 (πœ‘ β†’ 𝑃 ∈ 𝐡)
tglineelsb2.2 (πœ‘ β†’ 𝑄 ∈ 𝐡)
tglineelsb2.4 (πœ‘ β†’ 𝑃 β‰  𝑄)
Assertion
Ref Expression
tglinerflx1 (πœ‘ β†’ 𝑃 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx1
StepHypRef Expression
1 tglineelsb2.p . 2 𝐡 = (Baseβ€˜πΊ)
2 tglineelsb2.i . 2 𝐼 = (Itvβ€˜πΊ)
3 tglineelsb2.l . 2 𝐿 = (LineGβ€˜πΊ)
4 tglineelsb2.g . 2 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 tglineelsb2.1 . 2 (πœ‘ β†’ 𝑃 ∈ 𝐡)
6 tglineelsb2.2 . 2 (πœ‘ β†’ 𝑄 ∈ 𝐡)
7 tglineelsb2.4 . 2 (πœ‘ β†’ 𝑃 β‰  𝑄)
8 eqid 2733 . . 3 (distβ€˜πΊ) = (distβ€˜πΊ)
91, 8, 2, 4, 5, 6tgbtwntriv1 27742 . 2 (πœ‘ β†’ 𝑃 ∈ (𝑃𝐼𝑄))
101, 2, 3, 4, 5, 6, 5, 7, 9btwnlng1 27870 1 (πœ‘ β†’ 𝑃 ∈ (𝑃𝐿𝑄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiGcstrkg 27678  Itvcitv 27684  LineGclng 27685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-trkgc 27699  df-trkgb 27700  df-trkgcb 27701  df-trkg 27704
This theorem is referenced by:  tghilberti1  27888  tglnne0  27891  tglnpt2  27892  tglineneq  27895  coltr  27898  colline  27900  footexALT  27969  footexlem1  27970  footexlem2  27971  foot  27973  footne  27974  perprag  27977  colperp  27980  colperpexlem3  27983  mideulem2  27985  outpasch  28006  hlpasch  28007  lnopp2hpgb  28014  colopp  28020  lmieu  28035  lmimid  28045  hypcgrlem1  28050  hypcgrlem2  28051  trgcopyeulem  28056  tgasa1  28109
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