Theorem tgbtwntriv1 28887

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 28883	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
8	1, 2, 3, 4, 5, 6, 6, 7	tgbtwncom 28884	1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  distcds 17398  TarskiGcstrkg 28822  Itvcitv 28828

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6483  df-fv 6535  df-ov 7411  df-trkgc 28843  df-trkgb 28844  df-trkgcb 28845  df-trkg 28848

This theorem is used by:  tgldim0itv  28900  legtri3  28986  leg0  28988  legbtwn  28990  ncolne1  29026  tglnne  29029  tglinerflx1  29034  mirinv  29071  miriso  29075  colmid  29093  krippenlem  29095  colperpex  29142  outpasch  29166  hlpasch  29167  angmgmaddeu2  29313  angmgmaddeu3  29314  angmgmaddov2lem  29320  angmgmaddrid  29326