Theorem tgbtwnexch 28521

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 28511	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
11		tgbtwnexch.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
12	1, 2, 3, 4, 7, 6, 8, 11	tgbtwncom 28511	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
13	1, 2, 3, 4, 5, 8, 6, 7, 10, 12	tgbtwnexch2 28519	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
14	1, 2, 3, 4, 5, 6, 7, 13	tgbtwncom 28511	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476

This theorem is referenced by:  tgcgrxfr  28541  tgbtwnconn1lem1  28595  tgbtwnconn1lem3  28597  legtrd  28612  hltr  28633  hlbtwn  28634  tglineeltr  28654  miriso  28693  outpasch  28778