Theorem tgbtwntriv1 28771

Description: Betweenness always holds for the first endpoint. Theorem 3.3 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
tgbtwntriv1	⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv1

Step	Hyp	Ref	Expression
1		tkgeom.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. 2 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwntriv2.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
6		tgbtwntriv2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	1, 2, 3, 4, 5, 6	tgbtwntriv2 28767	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
8	1, 2, 3, 4, 5, 6, 6, 7	tgbtwncom 28768	1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  distcds 17325  TarskiGcstrkg 28707  Itvcitv 28713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  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-iota 6492  df-fv 6544  df-ov 7415  df-trkgc 28728  df-trkgb 28729  df-trkgcb 28730  df-trkg 28733

This theorem is used by:  tgldim0itv  28784  legtri3  28870  leg0  28872  legbtwn  28874  ncolne1  28909  tglnne  28912  tglinerflx1  28917  mirinv  28954  miriso  28958  colmid  28976  krippenlem  28978  colperpex  29025  outpasch  29048  hlpasch  29049