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Theorem cgracol 29148

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 2763	. . . . . . . . . 10 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
21	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane2 29133	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
22	21	necomd 3013	. . . . . . . . . . . . . . 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 29132	. . . . . . . . . . . . . . 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 28766	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
32	31	orcd 886	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
33	23, 25, 32	3jca 1146	. . . . . . . . . . . . 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 28766	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶))
42	41	olcd 887	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
43	34, 35, 42	3jca 1146	. . . . . . . . . . . . 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 28883	. . . . . . . . . . . . 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 28885	. . . . . . . . . 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 29147	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷((hlG‘𝐺)‘𝐸)𝐹)
50	1, 2, 20, 13, 17, 15, 5	ishlg 28883	. . . . . . . . 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 1162	. . . . . . 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 28766	. . . . . . . . 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 28766	. . . . . . . . 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 1104	. . . . 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 29142	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
74	1, 2, 20, 4, 12, 14, 16, 6, 8, 10, 73	cgrane1 29132	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
75	1, 72, 2, 4, 12, 14, 74, 16	tgellng 28831	. . . . 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 29146	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
89	1, 72, 2, 79, 80, 81, 82, 88	btwncolg3 28835	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
90	24	neneqd 2963	. . . . 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 28831	. . . 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 1104	. . 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ⟨“cs3 14884 Basecbs 17273 distcds 17323 TarskiGcstrkg 28705 Itvcitv 28711 LineGclng 28712 hlGchlg 28878 cgrAccgra 29127

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-dju 9892 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-n0 12509 df-xnn0 12582 df-z 12596 df-uz 12867 df-fz 13540 df-fzo 13688 df-hash 14372 df-word 14556 df-concat 14613 df-s1 14639 df-s2 14890 df-s3 14891 df-trkgc 28726 df-trkgb 28727 df-trkgcb 28728 df-trkg 28731 df-cgrg 28789 df-leg 28861 df-hlg 28879 df-cgra 29128

This theorem is used by: cgrancol 29149 tgasa1 29184

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