Theorem tgbtwntriv1 28581

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 28577	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
8	1, 2, 3, 4, 5, 6, 6, 7	tgbtwncom 28578	1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  distcds 17224  TarskiGcstrkg 28517  Itvcitv 28523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5231

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-trkgc 28538  df-trkgb 28539  df-trkgcb 28540  df-trkg 28543

This theorem is referenced by:  tgldim0itv  28594  legtri3  28680  leg0  28682  legbtwn  28684  ncolne1  28715  tglnne  28718  tglinerflx1  28723  mirinv  28756  miriso  28760  colmid  28778  krippenlem  28780  colperpex  28823  outpasch  28845  hlpasch  28846