Theorem tgbtwnintr

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 18-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 )
	tgbtwnintr.5	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐷 ) )
	tgbtwnintr.6	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐷 ) )
Assertion	tgbtwnintr	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )

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		tgbtwnintr.5	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐷 ) )
10		tgbtwnintr.6	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐷 ) )
11	4	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐺 ∈ TarskiG )
12	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐵 ∈ 𝑃 )
13		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ 𝑃 )
14		simprr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) )
15	1 2 3 11 12 13 14	axtgbtwnid	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐵 = 𝑥 )
16		simprl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) )
17	15 16	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
18	1 2 3 4 6 7 8 5 6 9 10	axtgpasch	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑃 ( 𝑥 ∈ ( 𝐴 𝐼 𝐶 ) ∧ 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ) )
19	17 18	r19.29a	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )