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Theorem cgracol 29096

Description: Angle congruence preserves colinearity. (Contributed by Thierry Arnoux, 9-Aug-2020.)

**Hypotheses**
Ref	Expression
cgracol.p	⊢ 𝑃 = (Base‘𝐺)
cgracol.i	⊢ 𝐼 = (Itv‘𝐺)
cgracol.m	⊢ − = (dist‘𝐺)
cgracol.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
cgracol.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
cgracol.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
cgracol.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
cgracol.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
cgracol.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
cgracol.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgracol.1	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
cgracol.l	⊢ 𝐿 = (LineG‘𝐺)
cgracol.2	⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))

**Assertion**
Ref	Expression
cgracol	⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

**Proof of Theorem cgracol**
Step	Hyp	Ref	Expression
1		cgracol.p	. . . . . . . . . 10 ⊢ 𝑃 = (Base‘𝐺)
2		cgracol.i	. . . . . . . . . 10 ⊢ 𝐼 = (Itv‘𝐺)
3		cgracol.m	. . . . . . . . . 10 ⊢ − = (dist‘𝐺)
4		cgracol.g	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG)
6		cgracol.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐴 ∈ 𝑃)
8		cgracol.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ 𝑃)
10		cgracol.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐶 ∈ 𝑃)
12		cgracol.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷 ∈ 𝑃)
14		cgracol.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐸 ∈ 𝑃)
16		cgracol.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
17	16	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐹 ∈ 𝑃)
18		cgracol.1	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
19	18	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
20		eqid 2769	. . . . . . . . . 10 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
21	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane2 29081	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
22	21	necomd 3019	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
23	22	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ≠ 𝐵)
24	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane1 29080	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
25	24	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴 ≠ 𝐵)
26	4	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
27	6	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
28	10	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ 𝑃)
29	8	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃)
30		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐴𝐼𝐵))
31	1, 3, 2, 26, 27, 28, 29, 30	tgbtwncom 28723	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
32	31	orcd 886	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
33	23, 25, 32	3jca 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
34	22	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ≠ 𝐵)
35	24	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ≠ 𝐵)
36	4	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG)
37	10	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃)
38	6	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃)
39	8	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃)
40		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵))
41	1, 3, 2, 36, 37, 38, 39, 40	tgbtwncom 28723	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶))
42	41	olcd 887	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
43	34, 35, 42	3jca 1144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
44	33, 43	jaodan 972	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
45	1, 2, 20, 10, 6, 8, 4	ishlg 28837	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶((hlG‘𝐺)‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))))
46	45	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶((hlG‘𝐺)‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))))
47	44, 46	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐶((hlG‘𝐺)‘𝐵)𝐴)
48	1, 2, 20, 11, 7, 9, 5, 47	hlcomd 28839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐴((hlG‘𝐺)‘𝐵)𝐶)
49	1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 48	cgrahl 29095	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷((hlG‘𝐺)‘𝐸)𝐹)
50	1, 2, 20, 13, 17, 15, 5	ishlg 28837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷((hlG‘𝐺)‘𝐸)𝐹 ↔ (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))))
51	49, 50	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))))
52	51	simp3d 1160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))
53	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐺 ∈ TarskiG)
54	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐸 ∈ 𝑃)
55	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ 𝑃)
56	16	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐹 ∈ 𝑃)
57		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐸𝐼𝐹))
58	1, 3, 2, 53, 54, 55, 56, 57	tgbtwncom 28723	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐹𝐼𝐸))
59	58	olcd 887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
60	4	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐺 ∈ TarskiG)
61	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐸 ∈ 𝑃)
62	16	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ 𝑃)
63	12	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐷 ∈ 𝑃)
64		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐸𝐼𝐷))
65	1, 3, 2, 60, 61, 62, 63, 64	tgbtwncom 28723	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐷𝐼𝐸))
66	65	orcd 886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
67	59, 66	jaodan 972	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
68	52, 67	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
69	68	orcd 886	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
70		df-3or 1102	. . . . 5 ⊢ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)) ↔ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
71	69, 70	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
72		cgracol.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
73	1, 2, 4, 20, 6, 8, 10, 12, 14, 16, 18	cgracom 29090	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
74	1, 2, 20, 4, 12, 14, 16, 6, 8, 10, 73	cgrane1 29080	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
75	1, 72, 2, 4, 12, 14, 74, 16	tgellng 28788	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ↔ (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹))))
76	75	adantr 485	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐿𝐸) ↔ (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹))))
77	71, 76	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐹 ∈ (𝐷𝐿𝐸))
78	77	orcd 886	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
79	4	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
80	12	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ 𝑃)
81	14	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ 𝑃)
82	16	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐹 ∈ 𝑃)
83	6	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
84	8	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ 𝑃)
85	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
86	18	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
87		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
88	1, 2, 3, 79, 83, 84, 85, 80, 81, 82, 86, 87	cgrabtwn 29094	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
89	1, 72, 2, 79, 80, 81, 82, 88	btwncolg3 28792	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
90	24	neneqd 2969	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
91		cgracol.2	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
92	91	orcomd 884	. . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 ∈ (𝐴𝐿𝐵)))
93	92	ord 877	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐶 ∈ (𝐴𝐿𝐵)))
94	90, 93	mpd 16	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵))
95	1, 72, 2, 4, 6, 8, 24, 10	tgellng 28788	. . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶))))
96	94, 95	mpbid 235	. . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
97		df-3or 1102	. . 3 ⊢ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)) ↔ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
98	96, 97	sylib 221	. 2 ⊢ (𝜑 → ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
99	78, 89, 98	mpjaodan 973	1 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1100 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5113 ‘cfv 6537 (class class class)co 7411 ⟨“cs3 14879 Basecbs 17269 distcds 17319 TarskiGcstrkg 28662 Itvcitv 28668 LineGclng 28669 hlGchlg 28835 cgrAccgra 29075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-oadd 8457 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-xnn0 12578 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683 df-hash 14367 df-word 14551 df-concat 14608 df-s1 14634 df-s2 14885 df-s3 14886 df-trkgc 28683 df-trkgb 28684 df-trkgcb 28685 df-trkg 28688 df-cgrg 28746 df-leg 28818 df-hlg 28836 df-cgra 29076

This theorem is referenced by: cgrancol 29097 tgasa1 29130

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