Theorem tgbtwnouttr 26858

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 2999	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11		tgbtwnouttr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12	1, 2, 3, 4, 6, 8, 5, 11	tgbtwncom 26849	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵))
13		tgbtwnouttr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
14	1, 2, 3, 4, 7, 6, 8, 13	tgbtwncom 26849	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
15	1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14	tgbtwnouttr2 26856	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
16	1, 2, 3, 4, 5, 6, 7, 15	tgbtwncom 26849	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814

This theorem is referenced by:  btwnhl  26975  tglineeltr  26992  outpasch  27116