Theorem tgbtwnexch 28721

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 28711	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐴))
11		tgbtwnexch.1	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
12	1, 2, 3, 4, 7, 6, 8, 11	tgbtwncom 28711	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
13	1, 2, 3, 4, 5, 8, 6, 7, 10, 12	tgbtwnexch2 28719	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
14	1, 2, 3, 4, 5, 6, 7, 13	tgbtwncom 28711	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  distcds 17307  TarskiGcstrkg 28650  Itvcitv 28656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-nul 5260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ov 7403  df-trkgc 28671  df-trkgb 28672  df-trkgcb 28673  df-trkg 28676

This theorem is referenced by:  tgcgrxfr  28741  tgbtwnconn1lem1  28795  tgbtwnconn1lem3  28797  legtrd  28812  hltr  28833  hlbtwn  28834  tglineeltr  28854  miriso  28897  outpasch  28982