Theorem tgbtwnouttr 28837

Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 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)
tgbtwnouttr.1	⊢ (𝜑 → 𝐵 ≠ 𝐶)
tgbtwnouttr.2	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnouttr.3	⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))

Assertion

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

Proof of Theorem tgbtwnouttr

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		tgbtwnouttr.1	. . . 4 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
10	9	necomd 3012	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11		tgbtwnouttr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12	1, 2, 3, 4, 6, 8, 5, 11	tgbtwncom 28828	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵))
13		tgbtwnouttr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
14	1, 2, 3, 4, 7, 6, 8, 13	tgbtwncom 28828	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
15	1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14	tgbtwnouttr2 28835	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
16	1, 2, 3, 4, 5, 6, 7, 15	tgbtwncom 28828	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  distcds 17355  TarskiGcstrkg 28766  Itvcitv 28772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545  df-ov 7419  df-trkgc 28787  df-trkgb 28788  df-trkgcb 28789  df-trkg 28792

This theorem is used by:  btwnhl  28957  tglineeltr  28976  outpasch  29110