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Theorem fvline 35676
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 2727 . . . . 5 [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear
2 fveq2 6891 . . . . . . . . 9 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32eleq2d 2814 . . . . . . . 8 (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
42eleq2d 2814 . . . . . . . 8 (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
53, 43anbi12d 1434 . . . . . . 7 (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)))
65anbi1d 629 . . . . . 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 3607 . . . . 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 703 . . . 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 1192 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10 simpr2 1193 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11 colinearex 35592 . . . . . . . 8 Colinear ∈ V
1211cnvex 7927 . . . . . . 7 Colinear ∈ V
13 ecexg 8722 . . . . . . 7 ( Colinear ∈ V → [⟨𝐴, 𝐵⟩] Colinear ∈ V)
1412, 13ax-mp 5 . . . . . 6 [⟨𝐴, 𝐵⟩] Colinear ∈ V
15 eleq1 2816 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16 neeq1 2998 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝑏𝐴𝑏))
1715, 163anbi13d 1435 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏)))
18 opeq1 4869 . . . . . . . . . . 11 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
1918eceq1d 8757 . . . . . . . . . 10 (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩] Colinear = [⟨𝐴, 𝑏⟩] Colinear )
2019eqeq2d 2738 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ))
2117, 20anbi12d 630 . . . . . . . 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 3173 . . . . . . 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 2816 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24 neeq2 2999 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2523, 243anbi23d 1436 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵)))
26 opeq2 4870 . . . . . . . . . . 11 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
2726eceq1d 8757 . . . . . . . . . 10 (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )
2827eqeq2d 2738 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ))
2925, 28anbi12d 630 . . . . . . . 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 3173 . . . . . . 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 2731 . . . . . . . . 9 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (𝑙 = [⟨𝐴, 𝐵⟩] Colinear ↔ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
3231anbi2d 628 . . . . . . . 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 3173 . . . . . . 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 7524 . . . . . 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 1447 . . . . 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 583 . . . 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 257 . . 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 7417 . . . 4 (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39 df-br 5143 . . . . . 6 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ Line)
40 df-line2 35669 . . . . . . 7 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
4140eleq2i 2820 . . . . . 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 275 . . . . 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 35674 . . . . . 6 Fun Line
44 funbrfv 6942 . . . . . 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 234 . . . 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, 46eqtrid 2779 . . 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 5460 . . . 4 𝐴, 𝐵⟩ ∈ V
50 dfec2 8721 . . . 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 3473 . . . . 5 𝑥 ∈ V
5349, 52brcnv 5879 . . . 4 (⟨𝐴, 𝐵 Colinear 𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩)
5453abbii 2797 . . 3 {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5551, 54eqtri 2755 . 2 [⟨𝐴, 𝐵⟩] Colinear = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5648, 55eqtrdi 2783 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  {cab 2704  wne 2935  wrex 3065  Vcvv 3469  cop 4630   class class class wbr 5142  ccnv 5671  Fun wfun 6536  cfv 6542  (class class class)co 7414  {coprab 7415  [cec 8716  cn 12234  𝔼cee 28686   Colinear ccolin 35569  Linecline2 35666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-1cn 11188  ax-addcl 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-ec 8720  df-nn 12235  df-colinear 35571  df-line2 35669
This theorem is referenced by:  liness  35677  fvline2  35678  ellines  35684
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