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Theorem tgbtwnexch 28506
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 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)
tgbtwnexch.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch.2 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch (𝜑𝐵 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch
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 tgbtwnexch.2 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
101, 2, 3, 4, 7, 8, 5, 9tgbtwncom 28496 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐴))
11 tgbtwnexch.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
121, 2, 3, 4, 7, 6, 8, 11tgbtwncom 28496 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
131, 2, 3, 4, 5, 8, 6, 7, 10, 12tgbtwnexch2 28504 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
141, 2, 3, 4, 5, 6, 7, 13tgbtwncom 28496 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461
This theorem is referenced by:  tgcgrxfr  28526  tgbtwnconn1lem1  28580  tgbtwnconn1lem3  28582  legtrd  28597  hltr  28618  hlbtwn  28619  tglineeltr  28639  miriso  28678  outpasch  28763
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