Theorem tgbtwnconn22

Description: Double connectivity law for betweenness. (Contributed by Thierry Arnoux, 1-Dec-2019)

	Ref	Expression
Hypotheses	tgbtwnconn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgbtwnconn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgbtwnconn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnconn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnconn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnconn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwnconn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgbtwnconn22.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
	tgbtwnconn22.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	tgbtwnconn22.2	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
	tgbtwnconn22.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
	tgbtwnconn22.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
	tgbtwnconn22.5	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐸 ) )
Assertion	tgbtwnconn22	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐷 𝐼 𝐸 ) )

Proof

Step	Hyp	Ref	Expression
1		tgbtwnconn.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgbtwnconn.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		tgbtwnconn.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
4		tgbtwnconn.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
5		tgbtwnconn.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
6		tgbtwnconn.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
7		tgbtwnconn.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
8		tgbtwnconn22.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
9		tgbtwnconn22.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
10		tgbtwnconn22.2	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
11		tgbtwnconn22.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
12		tgbtwnconn22.4	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
13		tgbtwnconn22.5	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐶 𝐼 𝐸 ) )
14		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
15	3	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐺 ∈ TarskiG )
16	7	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐷 ∈ 𝑃 )
17	6	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ∈ 𝑃 )
18	5	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐵 ∈ 𝑃 )
19	8	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐸 ∈ 𝑃 )
20	10	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ≠ 𝐵 )
21		simpr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) )
22	1 14 2 15 18 17 16 21	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐷 𝐼 𝐵 ) )
23	13	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐸 ) )
24	1 14 2 15 16 17 18 19 20 22 23	tgbtwnouttr2	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐵 ∈ ( 𝐷 𝐼 𝐸 ) )
25	3	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG )
26	7	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐷 ∈ 𝑃 )
27	5	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐵 ∈ 𝑃 )
28	8	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐸 ∈ 𝑃 )
29	6	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 )
30		simpr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) )
31	13	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐶 𝐼 𝐸 ) )
32	1 14 2 25 29 27 28 31	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐸 𝐼 𝐶 ) )
33	1 14 2 25 26 27 28 29 30 32	tgbtwnintr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐵 ∈ ( 𝐷 𝐼 𝐸 ) )
34	1 2 3 4 5 6 7 9 11 12	tgbtwnconn2	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) )
35	24 33 34	mpjaodan	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐷 𝐼 𝐸 ) )