Theorem tgbtwnouttr 28632

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 3002	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11		tgbtwnouttr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12	1, 2, 3, 4, 6, 8, 5, 11	tgbtwncom 28623	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵))
13		tgbtwnouttr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
14	1, 2, 3, 4, 7, 6, 8, 13	tgbtwncom 28623	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
15	1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14	tgbtwnouttr2 28630	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
16	1, 2, 3, 4, 5, 6, 7, 15	tgbtwncom 28623	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  distcds 17267  TarskiGcstrkg 28562  Itvcitv 28568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-nul 5246

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-iota 6462  df-fv 6514  df-ov 7384  df-trkgc 28583  df-trkgb 28584  df-trkgcb 28585  df-trkg 28588

This theorem is referenced by:  btwnhl  28749  tglineeltr  28766  outpasch  28890