Theorem tglinerflx1 28809

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 2763	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv1 28667	. 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 5, 7, 9	btwnlng1 28795	1 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ‘cfv 6521  (class class class)co 7396  Basecbs 17255  distcds 17305  TarskiGcstrkg 28603  Itvcitv 28609  LineGclng 28610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-iota 6477  df-fun 6523  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-trkgc 28624  df-trkgb 28625  df-trkgcb 28626  df-trkg 28629

This theorem is referenced by:  tghilberti1  28813  tglinesseq  28816  tglnne0  28817  tglineneq  28821  coltr  28824  colline  28826  tglnpt2  28829  footexALT  28898  footexlem1  28899  footexlem2  28900  foot  28902  footne  28903  perprag  28906  colperp  28909  colperpexlem3  28912  mideulem2  28914  outpasch  28935  hlpasch  28936  lnopp2hpgb  28943  colopp  28949  lmieu  28964  lmimid  28974  hypcgrlem1  28979  hypcgrlem2  28980  trgcopyeulem  28985  plngrotlem1  29001  plngrotlem2  29002  lnssplnglem  29005  lnssplng  29006  plng3p  29007  tgasa1  29059