MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnexch3 Structured version   Visualization version   GIF version

Theorem tgbtwnexch3 27265
Description: Exchange the first endpoint in betweenness. Left-hand side of Theorem 3.6 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwnintr.1 (𝜑𝐴𝑃)
tgbtwnintr.2 (𝜑𝐵𝑃)
tgbtwnintr.3 (𝜑𝐶𝑃)
tgbtwnintr.4 (𝜑𝐷𝑃)
tgbtwnexch3.5 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch3.6 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch3 (𝜑𝐶 ∈ (𝐵𝐼𝐷))

Proof of Theorem tgbtwnexch3
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 tgbtwnintr.2 . 2 (𝜑𝐵𝑃)
6 tgbtwnintr.3 . 2 (𝜑𝐶𝑃)
7 tgbtwnintr.4 . 2 (𝜑𝐷𝑃)
8 tgbtwnintr.1 . 2 (𝜑𝐴𝑃)
9 tgbtwnexch3.5 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
101, 2, 3, 4, 8, 5, 6, 9tgbtwncom 27259 . 2 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
11 tgbtwnexch3.6 . . 3 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
121, 2, 3, 4, 8, 6, 7, 11tgbtwncom 27259 . 2 (𝜑𝐶 ∈ (𝐷𝐼𝐴))
131, 2, 3, 4, 5, 6, 7, 8, 10, 12tgbtwnintr 27264 1 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351  Basecbs 17043  distcds 17102  TarskiGcstrkg 27198  Itvcitv 27204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-trkgc 27219  df-trkgb 27220  df-trkgcb 27221  df-trkg 27224
This theorem is referenced by:  tgbtwnouttr2  27266  tgifscgr  27279  tgcgrxfr  27289  tgbtwnconn1lem1  27343  tgbtwnconn1lem2  27344  tgbtwnconn1lem3  27345  tgbtwnconn2  27347  tgbtwnconn3  27348  btwnhl  27385  tglineeltr  27402  miriso  27441  krippenlem  27461  outpasch  27526  hlpasch  27527
  Copyright terms: Public domain W3C validator