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Theorem cgracol 28619

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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐺 ∈ TarskiG)
6		cgracol.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐴 ∈ 𝑃)
8		cgracol.b	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
9	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐵 ∈ 𝑃)
10		cgracol.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐶 ∈ 𝑃)
12		cgracol.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷 ∈ 𝑃)
14		cgracol.e	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐸 ∈ 𝑃)
16		cgracol.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐹 ∈ 𝑃)
18		cgracol.1	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
19	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
20		eqid 2727	. . . . . . . . . 10 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
21	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane2 28604	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
22	21	necomd 2991	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
23	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ≠ 𝐵)
24	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane1 28603	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
25	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴 ≠ 𝐵)
26	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
27	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃)
28	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ 𝑃)
29	8	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐵 ∈ 𝑃)
30		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐴𝐼𝐵))
31	1, 3, 2, 26, 27, 28, 29, 30	tgbtwncom 28279	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
32	31	orcd 872	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
33	23, 25, 32	3jca 1126	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
34	22	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ≠ 𝐵)
35	24	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ≠ 𝐵)
36	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐺 ∈ TarskiG)
37	10	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐶 ∈ 𝑃)
38	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ 𝑃)
39	8	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐵 ∈ 𝑃)
40		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐶𝐼𝐵))
41	1, 3, 2, 36, 37, 38, 39, 40	tgbtwncom 28279	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶))
42	41	olcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
43	34, 35, 42	3jca 1126	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
44	33, 43	jaodan 956	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
45	1, 2, 20, 10, 6, 8, 4	ishlg 28393	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶((hlG‘𝐺)‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))))
46	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶((hlG‘𝐺)‘𝐵)𝐴 ↔ (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))))
47	44, 46	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐶((hlG‘𝐺)‘𝐵)𝐴)
48	1, 2, 20, 11, 7, 9, 5, 47	hlcomd 28395	. . . . . . . . . 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 28618	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷((hlG‘𝐺)‘𝐸)𝐹)
50	1, 2, 20, 13, 17, 15, 5	ishlg 28393	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷((hlG‘𝐺)‘𝐸)𝐹 ↔ (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))))
51	49, 50	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))))
52	51	simp3d 1142	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))
53	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐺 ∈ TarskiG)
54	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐸 ∈ 𝑃)
55	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ 𝑃)
56	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐹 ∈ 𝑃)
57		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐸𝐼𝐹))
58	1, 3, 2, 53, 54, 55, 56, 57	tgbtwncom 28279	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐹𝐼𝐸))
59	58	olcd 873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
60	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐺 ∈ TarskiG)
61	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐸 ∈ 𝑃)
62	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ 𝑃)
63	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐷 ∈ 𝑃)
64		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐸𝐼𝐷))
65	1, 3, 2, 60, 61, 62, 63, 64	tgbtwncom 28279	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐷𝐼𝐸))
66	65	orcd 872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
67	59, 66	jaodan 956	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
68	52, 67	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
69	68	orcd 872	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
70		df-3or 1086	. . . . 5 ⊢ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)) ↔ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
71	69, 70	sylibr 233	. . . 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 28613	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
74	1, 2, 20, 4, 12, 14, 16, 6, 8, 10, 73	cgrane1 28603	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
75	1, 72, 2, 4, 12, 14, 74, 16	tgellng 28344	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ↔ (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹))))
76	75	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐿𝐸) ↔ (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹))))
77	71, 76	mpbird 257	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐹 ∈ (𝐷𝐿𝐸))
78	77	orcd 872	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
79	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
80	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ 𝑃)
81	14	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ 𝑃)
82	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐹 ∈ 𝑃)
83	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃)
84	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ 𝑃)
85	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃)
86	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
87		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
88	1, 2, 3, 79, 83, 84, 85, 80, 81, 82, 86, 87	cgrabtwn 28617	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
89	1, 72, 2, 79, 80, 81, 82, 88	btwncolg3 28348	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
90	24	neneqd 2940	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
91		cgracol.2	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
92	91	orcomd 870	. . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 ∈ (𝐴𝐿𝐵)))
93	92	ord 863	. . . . 5 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐶 ∈ (𝐴𝐿𝐵)))
94	90, 93	mpd 15	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐿𝐵))
95	1, 72, 2, 4, 6, 8, 24, 10	tgellng 28344	. . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶))))
96	94, 95	mpbid 231	. . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
97		df-3or 1086	. . 3 ⊢ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)) ↔ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
98	96, 97	sylib 217	. 2 ⊢ (𝜑 → ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
99	78, 89, 98	mpjaodan 957	1 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 ∨ w3o 1084 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ⟨“cs3 14817 Basecbs 17171 distcds 17233 TarskiGcstrkg 28218 Itvcitv 28224 LineGclng 28225 hlGchlg 28391 cgrAccgra 28598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9916 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-xnn0 12567 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-hash 14314 df-word 14489 df-concat 14545 df-s1 14570 df-s2 14823 df-s3 14824 df-trkgc 28239 df-trkgb 28240 df-trkgcb 28241 df-trkg 28244 df-cgrg 28302 df-leg 28374 df-hlg 28392 df-cgra 28599

This theorem is referenced by: cgrancol 28620 tgasa1 28649

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