Theorem tglinerflx2 28153

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
tglinerflx2	⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

Proof of Theorem tglinerflx2

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 2731	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv2 28006	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 6, 7, 9	btwnlng1 28138	1 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952  LineGclng 27953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-trkgc 27967  df-trkgcb 27969  df-trkg 27972

This theorem is referenced by:  tghilberti1  28156  tglnpt2  28160  colline  28168  footexALT  28237  footexlem2  28239  foot  28241  footne  28242  perprag  28245  colperpexlem3  28251  mideulem2  28253  opphllem  28254  opphllem5  28270  opphllem6  28271  opphl  28273  outpasch  28274  hlpasch  28275  lnopp2hpgb  28282  hypcgrlem1  28318  hypcgrlem2  28319  trgcopyeulem  28324  acopy  28352  acopyeu  28353  tgasa1  28377