Theorem tglinerflx2 28720

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 2737	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
9	1, 8, 2, 4, 5, 6	tgbtwntriv2 28573	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐼𝑄))
10	1, 2, 3, 4, 5, 6, 6, 7, 9	btwnlng1 28705	1 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6494  (class class class)co 7362  Basecbs 17174  distcds 17224  TarskiGcstrkg 28513  Itvcitv 28519  LineGclng 28520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-iota 6450  df-fun 6496  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-trkgc 28534  df-trkgcb 28536  df-trkg 28539

This theorem is referenced by:  tghilberti1  28723  tglnpt2  28727  colline  28735  footexALT  28804  footexlem2  28806  foot  28808  footne  28809  perprag  28812  colperpexlem3  28818  mideulem2  28820  opphllem  28821  opphllem5  28837  opphllem6  28838  opphl  28840  outpasch  28841  hlpasch  28842  lnopp2hpgb  28849  hypcgrlem1  28885  hypcgrlem2  28886  trgcopyeulem  28891  acopy  28919  acopyeu  28920  tgasa1  28944