Theorem tgbtwnouttr

Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 23-Mar-2019)

	Ref	Expression
Hypotheses	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 ∈ ( 𝐴 𝐼 𝐶 ) )
	tgbtwnouttr.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
Assertion	tgbtwnouttr	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
3		tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
5		tgbtwnintr.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwnintr.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgbtwnintr.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgbtwnintr.4	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
9		tgbtwnouttr.1	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
10		tgbtwnouttr.2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
11		tgbtwnouttr.3	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
12	9	necomd	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
13	1 2 3 4 6 7 8 11	tgbtwncom	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐷 𝐼 𝐵 ) )
14	1 2 3 4 5 6 7 10	tgbtwncom	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
15	1 2 3 4 8 7 6 5 12 13 14	tgbtwnouttr2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐷 𝐼 𝐴 ) )
16	1 2 3 4 8 6 5 15	tgbtwncom	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )