Theorem tglinerflx1 28882

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tglineelsb2.1	⊢ (𝜑 → 𝑃 ∈ 𝐵)
tglineelsb2.2	⊢ (𝜑 → 𝑄 ∈ 𝐵)
tglineelsb2.4	⊢ (𝜑 → 𝑃 ≠ 𝑄)

Assertion

Ref	Expression
tglinerflx1	⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx1

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tglineelsb2.1	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
6		tglineelsb2.2	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
7		tglineelsb2.4	. 2 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
8		eqid 2761	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv1 28736	. 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 5, 7, 9	btwnlng1 28868	1 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  distcds 17318  TarskiGcstrkg 28672  Itvcitv 28678  LineGclng 28679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-trkgc 28693  df-trkgb 28694  df-trkgcb 28695  df-trkg 28698

This theorem is referenced by:  tghilberti1  28886  tglinesseq  28889  tglnne0  28890  tglineneq  28894  coltr  28897  colline  28899  tglnpt2  28902  footexALT  28973  footexlem1  28974  footexlem2  28975  foot  28977  footne  28978  perprag  28982  colperp  28985  colperpexlem3  28988  mideulem2  28990  outpasch  29012  hlpasch  29013  lnopp2hpgb  29020  colopp  29026  plngrotlem1  29043  plngrotlem2  29044  lnssplnglem  29047  lnssplng  29048  mirplncl  29051  plng3p  29053  lmieu  29067  lmimid  29077  hypcgrlem1  29082  hypcgrlem2  29083  trgcopyeulem  29089  perpeq  29124  tgasa1  29148  dfprlng2  29170  prlngex  29174  prlngmolem1  29175  prlngmid2  29183