Theorem tgbtwnexch 28584

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 28574	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
11		tgbtwnexch.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
12	1, 2, 3, 4, 7, 6, 8, 11	tgbtwncom 28574	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
13	1, 2, 3, 4, 5, 8, 6, 7, 10, 12	tgbtwnexch2 28582	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
14	1, 2, 3, 4, 5, 6, 7, 13	tgbtwncom 28574	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  distcds 17220  TarskiGcstrkg 28513  Itvcitv 28519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-ov 7359  df-trkgc 28534  df-trkgb 28535  df-trkgcb 28536  df-trkg 28539

This theorem is referenced by:  tgcgrxfr  28604  tgbtwnconn1lem1  28658  tgbtwnconn1lem3  28660  legtrd  28675  hltr  28696  hlbtwn  28697  tglineeltr  28717  miriso  28756  outpasch  28841