Theorem tgbtwntriv1 28469

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 28465	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵𝐼𝐴))
8	1, 2, 3, 4, 5, 6, 6, 7	tgbtwncom 28466	1 ⊢ (𝜑 → 𝐴 ∈ (𝐴𝐼𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  distcds 17170  TarskiGcstrkg 28405  Itvcitv 28411

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 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-trkgc 28426  df-trkgb 28427  df-trkgcb 28428  df-trkg 28431

This theorem is referenced by:  tgldim0itv  28482  legtri3  28568  leg0  28570  legbtwn  28572  ncolne1  28603  tglnne  28606  tglinerflx1  28611  mirinv  28644  miriso  28648  colmid  28666  krippenlem  28668  colperpex  28711  outpasch  28733  hlpasch  28734