Theorem fvline 34084

Description: Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fvline	⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem fvline

Dummy variables 𝑎 𝑏 𝑙 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear
2		fveq2 6674	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
3	2	eleq2d 2818	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
4	2	eleq2d 2818	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
5	3, 4	3anbi12d 1438	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)))
6	5	anbi1d 633	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
7	6	rspcev 3526	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
8	1, 7	mpanr2 704	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
9		simpr1 1195	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10		simpr2 1196	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11		colinearex 34000	. . . . . . . 8 ⊢ Colinear ∈ V
12	11	cnvex 7656	. . . . . . 7 ⊢ ◡ Colinear ∈ V
13		ecexg 8324	. . . . . . 7 ⊢ (◡ Colinear ∈ V → [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V
15		eleq1 2820	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16		neeq1 2996	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
17	15, 16	3anbi13d 1439	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏)))
18		opeq1 4759	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
19	18	eceq1d 8359	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝑏⟩]◡ Colinear )
20	19	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ))
21	17, 20	anbi12d 634	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear )))
22	21	rexbidv 3207	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear )))
23		eleq1 2820	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24		neeq2 2997	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
25	23, 24	3anbi23d 1440	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵)))
26		opeq2 4761	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
27	26	eceq1d 8359	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )
28	27	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ))
29	25, 28	anbi12d 634	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear )))
30	29	rexbidv 3207	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear )))
31		eqeq1 2742	. . . . . . . . 9 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ↔ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
32	31	anbi2d 632	. . . . . . . 8 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
33	32	rexbidv 3207	. . . . . . 7 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
34	22, 30, 33	eloprabg 7276	. . . . . 6 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
35	14, 34	mp3an3 1451	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
36	9, 10, 35	syl2anc 587	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ∧ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )))
37	8, 36	mpbird 260	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
38		df-ov 7173	. . . 4 ⊢ (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39		df-br 5031	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ Line)
40		df-line2 34077	. . . . . . 7 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
41	40	eleq2i 2824	. . . . . 6 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ Line ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
42	39, 41	bitri 278	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )})
43		funline 34082	. . . . . 6 ⊢ Fun Line
44		funbrfv 6720	. . . . . 6 ⊢ (Fun Line → (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear ))
45	43, 44	ax-mp 5	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear )
46	42, 45	sylbir 238	. . . 4 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩]◡ Colinear )
47	38, 46	syl5eq 2785	. . 3 ⊢ (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )} → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩]◡ Colinear )
48	37, 47	syl 17	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩]◡ Colinear )
49		opex 5322	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
50		dfec2 8323	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥})
51	49, 50	ax-mp 5	. . 3 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥}
52		vex 3402	. . . . 5 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5725	. . . 4 ⊢ (⟨𝐴, 𝐵⟩◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝐴, 𝐵⟩)
54	53	abbii 2803	. . 3 ⊢ {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
55	51, 54	eqtri 2761	. 2 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
56	48, 55	eqtrdi 2789	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  {cab 2716   ≠ wne 2934  ∃wrex 3054  Vcvv 3398  ⟨cop 4522   class class class wbr 5030  ^◡ccnv 5524  Fun wfun 6333  ‘cfv 6339  (class class class)co 7170  {coprab 7171  [cec 8318  ℕcn 11716  𝔼cee 26834   Colinear ccolin 33977  Linecline2 34074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-1cn 10673  ax-addcl 10675

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-ec 8322  df-nn 11717  df-colinear 33979  df-line2 34077

This theorem is referenced by:  liness  34085  fvline2  34086  ellines  34092