Theorem tgbtwnouttr 26588

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 2996	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11		tgbtwnouttr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12	1, 2, 3, 4, 6, 8, 5, 11	tgbtwncom 26579	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵))
13		tgbtwnouttr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
14	1, 2, 3, 4, 7, 6, 8, 13	tgbtwncom 26579	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
15	1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14	tgbtwnouttr2 26586	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
16	1, 2, 3, 4, 5, 6, 7, 15	tgbtwncom 26579	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  distcds 16811  TarskiGcstrkg 26521  Itvcitv 26527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-trkgc 26539  df-trkgb 26540  df-trkgcb 26541  df-trkg 26544

This theorem is referenced by:  btwnhl  26705  tglineeltr  26722  outpasch  26846