Theorem tgbtwnexch 25817

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 25807	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
11		tgbtwnexch.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
12	1, 2, 3, 4, 7, 6, 8, 11	tgbtwncom 25807	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
13	1, 2, 3, 4, 5, 8, 6, 7, 10, 12	tgbtwnexch2 25815	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
14	1, 2, 3, 4, 5, 6, 7, 13	tgbtwncom 25807	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  distcds 16321  TarskiGcstrkg 25749  Itvcitv 25755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-trkgc 25767  df-trkgb 25768  df-trkgcb 25769  df-trkg 25772

This theorem is referenced by:  tgcgrxfr  25837  tgbtwnconn1lem1  25891  tgbtwnconn1lem3  25893  legtrd  25908  hltr  25929  hlbtwn  25930  tglineeltr  25950  miriso  25989  outpasch  26071