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Theorem tgbtwntriv1 25099
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 25095 . 2 (𝜑𝐴 ∈ (𝐵𝐼𝐴))
81, 2, 3, 4, 5, 6, 6, 7tgbtwncom 25096 1 (𝜑𝐴 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  cfv 5786  (class class class)co 6523  Basecbs 15637  distcds 15719  TarskiGcstrkg 25042  Itvcitv 25048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-nul 4708
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-ov 6526  df-trkgc 25060  df-trkgb 25061  df-trkgcb 25062  df-trkg 25065
This theorem is referenced by:  tgldim0itv  25112  legtri3  25199  leg0  25201  legbtwn  25203  ncolne1  25234  tglnne  25237  tglinerflx1  25242  mirinv  25275  miriso  25279  colmid  25297  krippenlem  25299  colperpex  25339  outpasch  25361  hlpasch  25362
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