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Theorem cgracol 28808

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 2729	. . . . . . . . . 10 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
21	1, 2, 20, 4, 6, 8, 10, 12, 14, 16, 18	cgrane2 28793	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
22	21	necomd 2980	. . . . . . . . . . . . . . 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 28792	. . . . . . . . . . . . . . 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 28468	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → 𝐶 ∈ (𝐵𝐼𝐴))
32	31	orcd 873	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
33	23, 25, 32	3jca 1128	. . . . . . . . . . . . 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 28468	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → 𝐴 ∈ (𝐵𝐼𝐶))
42	41	olcd 874	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
43	34, 35, 42	3jca 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝐶𝐼𝐵)) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
44	33, 43	jaodan 959	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐶 ≠ 𝐵 ∧ 𝐴 ≠ 𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶))))
45	1, 2, 20, 10, 6, 8, 4	ishlg 28582	. . . . . . . . . . . . 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 28584	. . . . . . . . . 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 28807	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → 𝐷((hlG‘𝐺)‘𝐸)𝐹)
50	1, 2, 20, 13, 17, 15, 5	ishlg 28582	. . . . . . . . 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 1144	. . . . . . 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 28468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐸𝐼𝐹)) → 𝐷 ∈ (𝐹𝐼𝐸))
59	58	olcd 874	. . . . . . . 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 28468	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → 𝐹 ∈ (𝐷𝐼𝐸))
66	65	orcd 873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ (𝐸𝐼𝐷)) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
67	59, 66	jaodan 959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐷 ∈ (𝐸𝐼𝐹) ∨ 𝐹 ∈ (𝐸𝐼𝐷))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
68	52, 67	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → (𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)))
69	68	orcd 873	. . . . 5 ⊢ ((𝜑 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐴 ∈ (𝐶𝐼𝐵))) → ((𝐹 ∈ (𝐷𝐼𝐸) ∨ 𝐷 ∈ (𝐹𝐼𝐸)) ∨ 𝐸 ∈ (𝐷𝐼𝐹)))
70		df-3or 1087	. . . . 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 28802	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
74	1, 2, 20, 4, 12, 14, 16, 6, 8, 10, 73	cgrane1 28792	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
75	1, 72, 2, 4, 12, 14, 74, 16	tgellng 28533	. . . . 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 873	. 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 28806	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → 𝐸 ∈ (𝐷𝐼𝐹))
89	1, 72, 2, 79, 80, 81, 82, 88	btwncolg3 28537	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝐴𝐼𝐶)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
90	24	neneqd 2930	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
91		cgracol.2	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
92	91	orcomd 871	. . . . . 6 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐶 ∈ (𝐴𝐿𝐵)))
93	92	ord 864	. . . . 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 28533	. . . 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 1087	. . 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 960	1 ⊢ (𝜑 → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 ⟨“cs3 14784 Basecbs 17155 distcds 17205 TarskiGcstrkg 28407 Itvcitv 28413 LineGclng 28414 hlGchlg 28580 cgrAccgra 28787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-concat 14512 df-s1 14537 df-s2 14790 df-s3 14791 df-trkgc 28428 df-trkgb 28429 df-trkgcb 28430 df-trkg 28433 df-cgrg 28491 df-leg 28563 df-hlg 28581 df-cgra 28788

This theorem is referenced by: cgrancol 28809 tgasa1 28838

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