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Theorem tgbtwntriv1 25380
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 25376 . 2 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 25377 1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  cfv 5886  (class class class)co 6647  Basecbs 15851  distcds 15944  TarskiGcstrkg 25323  Itvcitv 25329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-trkgc 25341  df-trkgb 25342  df-trkgcb 25343  df-trkg 25346
This theorem is referenced by:  tgldim0itv  25393  legtri3  25479  leg0  25481  legbtwn  25483  ncolne1  25514  tglnne  25517  tglinerflx1  25522  mirinv  25555  miriso  25559  colmid  25577  krippenlem  25579  colperpex  25619  outpasch  25641  hlpasch  25642
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