Theorem tgbtwnouttr 28553

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 2988	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11		tgbtwnouttr.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐷))
12	1, 2, 3, 4, 6, 8, 5, 11	tgbtwncom 28544	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐷𝐼𝐵))
13		tgbtwnouttr.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
14	1, 2, 3, 4, 7, 6, 8, 13	tgbtwncom 28544	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))
15	1, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14	tgbtwnouttr2 28551	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐴))
16	1, 2, 3, 4, 5, 6, 7, 15	tgbtwncom 28544	1 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ‘cfv 6490  (class class class)co 7358  Basecbs 17137  distcds 17187  TarskiGcstrkg 28483  Itvcitv 28489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6446  df-fv 6498  df-ov 7361  df-trkgc 28504  df-trkgb 28505  df-trkgcb 28506  df-trkg 28509

This theorem is referenced by:  btwnhl  28670  tglineeltr  28687  outpasch  28811