Theorem fvline 36495

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 2763	. . . . 5 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear
2		fveq2 6868	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
3	2	eleq2d 2849	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
4	2	eleq2d 2849	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
5	3, 4	3anbi12d 1459	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)))
6	5	anbi1d 640	. . . . . 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 3582	. . . . 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 714	. . . 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 1209	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10		simpr2 1210	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11		colinearex 36411	. . . . . . . 8 ⊢ Colinear ∈ V
12	11	cnvex 7907	. . . . . . 7 ⊢ ◡ Colinear ∈ V
13		ecexg 8683	. . . . . . 7 ⊢ (◡ Colinear ∈ V → [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V)
14	12, 13	ax-mp 5	. . . . . 6 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear ∈ V
15		eleq1 2851	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16		neeq1 3020	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏))
17	15, 16	3anbi13d 1460	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏)))
18		opeq1 4832	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
19	18	eceq1d 8720	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝑏⟩]◡ Colinear )
20	19	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ))
21	17, 20	anbi12d 641	. . . . . . . 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 3187	. . . . . . 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 2851	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24		neeq2 3021	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵))
25	23, 24	3anbi23d 1461	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴 ≠ 𝐵)))
26		opeq2 4833	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
27	26	eceq1d 8720	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear )
28	27	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩]◡ Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ))
29	25, 28	anbi12d 641	. . . . . . . 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 3187	. . . . . . 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 2767	. . . . . . . . 9 ⊢ (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear → (𝑙 = [⟨𝐴, 𝐵⟩]◡ Colinear ↔ [⟨𝐴, 𝐵⟩]◡ Colinear = [⟨𝐴, 𝐵⟩]◡ Colinear ))
32	31	anbi2d 639	. . . . . . . 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 3187	. . . . . . 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 7507	. . . . . 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 1472	. . . . 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 593	. . . 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 259	. . 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 7400	. . . 4 ⊢ (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39		df-br 5102	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩]◡ Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩]◡ Colinear ⟩ ∈ Line)
40		df-line2 36488	. . . . . . 7 ⊢ Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎 ≠ 𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩]◡ Colinear )}
41	40	eleq2i 2855	. . . . . 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 277	. . . . 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 36493	. . . . . 6 ⊢ Fun Line
44		funbrfv 6916	. . . . . 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 237	. . . 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	eqtrid 2810	. . 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 5432	. . . 4 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
50		dfec2 8682	. . . 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 3459	. . . . 5 ⊢ 𝑥 ∈ V
53	49, 52	brcnv 5855	. . . 4 ⊢ (⟨𝐴, 𝐵⟩◡ Colinear 𝑥 ↔ 𝑥 Colinear ⟨𝐴, 𝐵⟩)
54	53	abbii 2830	. . 3 ⊢ {𝑥 ∣ ⟨𝐴, 𝐵⟩◡ Colinear 𝑥} = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
55	51, 54	eqtri 2786	. 2 ⊢ [⟨𝐴, 𝐵⟩]◡ Colinear = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩}
56	48, 55	eqtrdi 2814	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵)) → (𝐴Line𝐵) = {𝑥 ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∃wrex 3087  Vcvv 3455  ⟨cop 4589   class class class wbr 5101  ^◡ccnv 5647  Fun wfun 6516  ‘cfv 6522  (class class class)co 7397  {coprab 7398  [cec 8677  ℕcn 12211  𝔼cee 29089   Colinear ccolin 36388  Linecline2 36485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719  ax-cnex 11130  ax-1cn 11132  ax-addcl 11134

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-ec 8681  df-nn 12212  df-colinear 36390  df-line2 36488

This theorem is referenced by:  liness  36496  fvline2  36497  ellines  36503