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

Theorem tgbtwnouttr 25810
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 23-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)
tgbtwnouttr.1 (𝜑𝐵𝐶)
tgbtwnouttr.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnouttr.3 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Assertion
Ref Expression
tgbtwnouttr (𝜑𝐵 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnouttr
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.4 . 2 (𝜑𝐷𝑃)
6 tgbtwnintr.2 . 2 (𝜑𝐵𝑃)
7 tgbtwnintr.1 . 2 (𝜑𝐴𝑃)
8 tgbtwnintr.3 . . 3 (𝜑𝐶𝑃)
9 tgbtwnouttr.1 . . . 4 (𝜑𝐵𝐶)
109necomd 3055 . . 3 (𝜑𝐶𝐵)
11 tgbtwnouttr.3 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
121, 2, 3, 4, 6, 8, 5, 11tgbtwncom 25801 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐵))
13 tgbtwnouttr.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
141, 2, 3, 4, 7, 6, 8, 13tgbtwncom 25801 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
151, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14tgbtwnouttr2 25808 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
161, 2, 3, 4, 5, 6, 7, 15tgbtwncom 25801 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  wne 3000  cfv 6124  (class class class)co 6906  Basecbs 16223  distcds 16315  TarskiGcstrkg 25743  Itvcitv 25749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-nul 5014
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-iota 6087  df-fv 6132  df-ov 6909  df-trkgc 25761  df-trkgb 25762  df-trkgcb 25763  df-trkg 25766
This theorem is referenced by:  btwnhl  25927  tglineeltr  25944  outpasch  26065
  Copyright terms: Public domain W3C validator