Theorem tgbtwnexch 28733

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwnintr.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwnintr.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwnintr.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwnintr.4	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgbtwnexch.1	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch.2	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))

Assertion

Ref	Expression
tgbtwnexch	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		tgbtwnintr.4	. 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
6		tgbtwnintr.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		tgbtwnintr.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		tgbtwnintr.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
9		tgbtwnexch.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷))
10	1, 2, 3, 4, 7, 8, 5, 9	tgbtwncom 28723	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
11		tgbtwnexch.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
12	1, 2, 3, 4, 7, 6, 8, 11	tgbtwncom 28723	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
13	1, 2, 3, 4, 5, 8, 6, 7, 10, 12	tgbtwnexch2 28731	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
14	1, 2, 3, 4, 5, 6, 7, 13	tgbtwncom 28723	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  distcds 17319  TarskiGcstrkg 28662  Itvcitv 28668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6493  df-fv 6545  df-ov 7414  df-trkgc 28683  df-trkgb 28684  df-trkgcb 28685  df-trkg 28688

This theorem is referenced by:  tgcgrxfr  28753  tgbtwnconn1lem1  28807  tgbtwnconn1lem3  28809  legtrd  28824  hltr  28845  hlbtwn  28846  tglineeltr  28866  miriso  28909  outpasch  28996