Theorem tgbtwncom

Description: Betweenness commutes. Theorem 3.2 of Schwabhauser p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019)

	Ref	Expression
Hypotheses	tkgeom.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tkgeom.d	⊢ − = ( dist ‘ 𝐺 )
	tkgeom.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tkgeom.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwncom.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwncom.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
Assertion	tgbtwncom	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )

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		tgbtwntriv2.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		tgbtwntriv2.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		tgbtwncom.3	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		tgbtwncom.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
9	4	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝐺 ∈ TarskiG )
10	6	ad2antrr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝐵 ∈ 𝑃 )
11		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝑥 ∈ 𝑃 )
12		simprl	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) )
13	1 2 3 9 10 11 12	axtgbtwnid	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝐵 = 𝑥 )
14		simprr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) )
15	13 14	eqeltrd	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑃 ) ∧ ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )
16	1 2 3 4 6 7	tgbtwntriv2	⊢ ( 𝜑 → 𝐶 ∈ ( 𝐵 𝐼 𝐶 ) )
17	1 2 3 4 5 6 7 6 7 8 16	axtgpasch	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝑃 ( 𝑥 ∈ ( 𝐵 𝐼 𝐵 ) ∧ 𝑥 ∈ ( 𝐶 𝐼 𝐴 ) ) )
18	15 17	r19.29a	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) )