MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwntriv1 Structured version   Visualization version   GIF version

Theorem tgbtwntriv1 26852
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 (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv1
StepHypRef 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 (𝜑𝐺 ∈ TarskiG)
5 tgbtwntriv2.2 . 2 (𝜑𝐵𝑃)
6 tgbtwntriv2.1 . 2 (𝜑𝐴𝑃)
71, 2, 3, 4, 5, 6tgbtwntriv2 26848 . 2 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 26849 1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814
This theorem is referenced by:  tgldim0itv  26865  legtri3  26951  leg0  26953  legbtwn  26955  ncolne1  26986  tglnne  26989  tglinerflx1  26994  mirinv  27027  miriso  27031  colmid  27049  krippenlem  27051  colperpex  27094  outpasch  27116  hlpasch  27117
  Copyright terms: Public domain W3C validator