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Theorem cgracol 28904

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 2737	. . . . . . . . . 10 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
21	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane2 28889	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
22	21	necomd 2988	. . . . . . . . . . . . . . 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 28888	. . . . . . . . . . . . . . 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 28564	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
32	31	orcd 874	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
33	23, 25, 32	3jca 1129	. . . . . . . . . . . . 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 28564	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶))
42	41	olcd 875	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
43	34, 35, 42	3jca 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
44	33, 43	jaodan 960	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
45	1, 2, 20, 10, 6, 8, 4	ishlg 28678	. . . . . . . . . . . . 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 28680	. . . . . . . . . 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 28903	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷((hlG‘𝐺)‘𝐸)𝐹)
50	1, 2, 20, 13, 17, 15, 5	ishlg 28678	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷((hlG‘𝐺)‘𝐸)𝐹 ↔ (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷)))))
51	49, 50	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐷 ≠ 𝐸 ∧ 𝐹 ≠ 𝐸 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))))
52	51	simp3d 1145	. . . . . . 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 28564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐹𝐼𝐸))
59	58	olcd 875	. . . . . . . 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 28564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐷𝐼𝐸))
66	65	orcd 874	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
67	59, 66	jaodan 960	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
68	52, 67	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
69	68	orcd 874	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
70		df-3or 1088	. . . . 5 ⊢ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸) ∨ 𝐸 ∈ (𝐷𝐼𝐹)) ↔ ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
71	69, 70	sylibr 234	. . . 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 28898	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
74	1, 2, 20, 4, 12, 14, 16, 6, 8, 10, 73	cgrane1 28888	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
75	1, 72, 2, 4, 12, 14, 74, 16	tgellng 28629	. . . . 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 874	. 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 28902	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
89	1, 72, 2, 79, 80, 81, 82, 88	btwncolg3 28633	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
90	24	neneqd 2938	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
91		cgracol.2	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
92	91	orcomd 872	. . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 ∈ (𝐴𝐿𝐵)))
93	92	ord 865	. . . . 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 28629	. . . 4 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ↔ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶))))
96	94, 95	mpbid 232	. . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
97		df-3or 1088	. . 3 ⊢ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)) ↔ ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
98	96, 97	sylib 218	. 2 ⊢ (𝜑 → ((𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵)) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
99	78, 89, 98	mpjaodan 961	1 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∨ w3o 1086 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 ⟨“cs3 14769 Basecbs 17140 distcds 17190 TarskiGcstrkg 28503 Itvcitv 28509 LineGclng 28510 hlGchlg 28676 cgrAccgra 28883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-xnn0 12479 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-concat 14498 df-s1 14524 df-s2 14775 df-s3 14776 df-trkgc 28524 df-trkgb 28525 df-trkgcb 28526 df-trkg 28529 df-cgrg 28587 df-leg 28659 df-hlg 28677 df-cgra 28884

This theorem is referenced by: cgrancol 28905 tgasa1 28934

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