Theorem tgbtwnexch

Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 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 )
	tgbtwnexch.1	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
	tgbtwnexch.2	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
Assertion	tgbtwnexch	⊢ ( 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		tgbtwnexch.1	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
10		tgbtwnexch.2	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
11	1 2 3 4 5 7 8 10	tgbtwncom	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐷 𝐼 𝐴 ) )
12	1 2 3 4 5 6 7 9	tgbtwncom	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
13	1 2 3 4 8 7 6 5 11 12	tgbtwnexch2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐷 𝐼 𝐴 ) )
14	1 2 3 4 8 6 5 13	tgbtwncom	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )