Theorem tgbtwnconnln1

Description: Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-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 )
	tgbtwnconnln1.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	tgbtwnconnln1.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	tgbtwnconnln1.2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
	tgbtwnconnln1.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
Assertion	tgbtwnconnln1	⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) )

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		tgbtwnconnln1.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
9		tgbtwnconnln1.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
10		tgbtwnconnln1.2	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) )
11		tgbtwnconnln1.3	⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) )
12	3	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐺 ∈ TarskiG )
13	6	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐶 ∈ 𝑃 )
14	7	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐷 ∈ 𝑃 )
15	4	adantr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐴 ∈ 𝑃 )
16		simpr	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) )
17	1 8 2 12 13 14 15 16	btwncolg2	⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) → ( 𝐴 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) )
18	3	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG )
19	6	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 )
20	7	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ 𝑃 )
21	4	adantr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐴 ∈ 𝑃 )
22		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
23		simpr	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) )
24	1 22 2 18 21 20 19 23	tgbtwncom	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐶 𝐼 𝐴 ) )
25	1 8 2 18 19 20 21 24	btwncolg3	⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) → ( 𝐴 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) )
26	1 2 3 4 5 6 7 9 10 11	tgbtwnconn1	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐴 𝐼 𝐶 ) ) )
27	17 25 26	mpjaodan	⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) )