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Theorem fvline 32578
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) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥𝑥 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.
StepHypRef Expression
1 eqid 2760 . . . . 5 [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear
2 fveq2 6353 . . . . . . . . 9 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32eleq2d 2825 . . . . . . . 8 (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
42eleq2d 2825 . . . . . . . 8 (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
53, 43anbi12d 1549 . . . . . . 7 (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)))
65anbi1d 743 . . . . . 6 (𝑛 = 𝑁 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
76rspcev 3449 . . . . 5 ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
81, 7mpanr2 722 . . . 4 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
9 simpr1 1234 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10 simpr2 1236 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11 colinearex 32494 . . . . . . . 8 Colinear ∈ V
1211cnvex 7279 . . . . . . 7 Colinear ∈ V
13 ecexg 7917 . . . . . . 7 ( Colinear ∈ V → [⟨𝐴, 𝐵⟩] Colinear ∈ V)
1412, 13ax-mp 5 . . . . . 6 [⟨𝐴, 𝐵⟩] Colinear ∈ V
15 eleq1 2827 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16 neeq1 2994 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝑏𝐴𝑏))
1715, 163anbi13d 1550 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏)))
18 opeq1 4553 . . . . . . . . . . 11 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
1918eceq1d 7952 . . . . . . . . . 10 (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩] Colinear = [⟨𝐴, 𝑏⟩] Colinear )
2019eqeq2d 2770 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ))
2117, 20anbi12d 749 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear )))
2221rexbidv 3190 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear )))
23 eleq1 2827 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24 neeq2 2995 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2523, 243anbi23d 1551 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵)))
26 opeq2 4554 . . . . . . . . . . 11 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
2726eceq1d 7952 . . . . . . . . . 10 (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )
2827eqeq2d 2770 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ))
2925, 28anbi12d 749 . . . . . . . 8 (𝑏 = 𝐵 → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear )))
3029rexbidv 3190 . . . . . . 7 (𝑏 = 𝐵 → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ∧ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear )))
31 eqeq1 2764 . . . . . . . . 9 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (𝑙 = [⟨𝐴, 𝐵⟩] Colinear ↔ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
3231anbi2d 742 . . . . . . . 8 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ) ↔ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
3332rexbidv 3190 . . . . . . 7 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ) ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
3422, 30, 33eloprabg 6914 . . . . . 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) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
3514, 34mp3an3 1562 . . . . 5 ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
369, 10, 35syl2anc 696 . . . 4 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )} ↔ ∃𝑛 ∈ ℕ ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ∧ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )))
378, 36mpbird 247 . . 3 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )})
38 df-ov 6817 . . . 4 (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39 df-br 4805 . . . . . 6 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ Line)
40 df-line2 32571 . . . . . . 7 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
4140eleq2i 2831 . . . . . 6 (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ Line ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )})
4239, 41bitri 264 . . . . 5 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )})
43 funline 32576 . . . . . 6 Fun Line
44 funbrfv 6396 . . . . . 6 (Fun Line → (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩] Colinear ))
4543, 44ax-mp 5 . . . . 5 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩] Colinear )
4642, 45sylbir 225 . . . 4 (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )} → (Line‘⟨𝐴, 𝐵⟩) = [⟨𝐴, 𝐵⟩] Colinear )
4738, 46syl5eq 2806 . . 3 (⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )} → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩] Colinear )
4837, 47syl 17 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = [⟨𝐴, 𝐵⟩] Colinear )
49 opex 5081 . . . 4 𝐴, 𝐵⟩ ∈ V
50 dfec2 7916 . . . 4 (⟨𝐴, 𝐵⟩ ∈ V → [⟨𝐴, 𝐵⟩] Colinear = {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥})
5149, 50ax-mp 5 . . 3 [⟨𝐴, 𝐵⟩] Colinear = {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥}
52 vex 3343 . . . . 5 𝑥 ∈ V
5349, 52brcnv 5460 . . . 4 (⟨𝐴, 𝐵 Colinear 𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩)
5453abbii 2877 . . 3 {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5551, 54eqtri 2782 . 2 [⟨𝐴, 𝐵⟩] Colinear = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5648, 55syl6eq 2810 1 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wne 2932  wrex 3051  Vcvv 3340  cop 4327   class class class wbr 4804  ccnv 5265  Fun wfun 6043  cfv 6049  (class class class)co 6814  {coprab 6815  [cec 7911  cn 11232  𝔼cee 25988   Colinear ccolin 32471  Linecline2 32568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-i2m1 10216  ax-1ne0 10217  ax-rrecex 10220  ax-cnre 10221
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-ec 7915  df-nn 11233  df-colinear 32473  df-line2 32571
This theorem is referenced by:  liness  32579  fvline2  32580  ellines  32586
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