Theorem tglinerflx2 28708

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 2736	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv2 28561	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 6, 7, 9	btwnlng1 28693	1 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  distcds 17188  TarskiGcstrkg 28501  Itvcitv 28507  LineGclng 28508

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-trkgc 28522  df-trkgcb 28524  df-trkg 28527

This theorem is referenced by:  tghilberti1  28711  tglnpt2  28715  colline  28723  footexALT  28792  footexlem2  28794  foot  28796  footne  28797  perprag  28800  colperpexlem3  28806  mideulem2  28808  opphllem  28809  opphllem5  28825  opphllem6  28826  opphl  28828  outpasch  28829  hlpasch  28830  lnopp2hpgb  28837  hypcgrlem1  28873  hypcgrlem2  28874  trgcopyeulem  28879  acopy  28907  acopyeu  28908  tgasa1  28932