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

Theorem tgbtwnouttr 25613
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 2998 . . 3 (𝜑𝐶𝐵)
11 tgbtwnouttr.3 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
121, 2, 3, 4, 6, 8, 5, 11tgbtwncom 25604 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐵))
13 tgbtwnouttr.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
141, 2, 3, 4, 7, 6, 8, 13tgbtwncom 25604 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
151, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14tgbtwnouttr2 25611 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
161, 2, 3, 4, 5, 6, 7, 15tgbtwncom 25604 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wne 2943  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  btwnhl  25730  tglineeltr  25747  outpasch  25868
  Copyright terms: Public domain W3C validator