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Theorem tgbtwnouttr 26210
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 3068 . . 3 (𝜑𝐶𝐵)
11 tgbtwnouttr.3 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
121, 2, 3, 4, 6, 8, 5, 11tgbtwncom 26201 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐵))
13 tgbtwnouttr.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
141, 2, 3, 4, 7, 6, 8, 13tgbtwncom 26201 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
151, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14tgbtwnouttr2 26208 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
161, 2, 3, 4, 5, 6, 7, 15tgbtwncom 26201 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wne 3013  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  btwnhl  26327  tglineeltr  26344  outpasch  26468
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