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Theorem fvline 31946
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 2621 . . . . 5 [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear
2 fveq2 6158 . . . . . . . . 9 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32eleq2d 2684 . . . . . . . 8 (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
42eleq2d 2684 . . . . . . . 8 (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
53, 43anbi12d 1397 . . . . . . 7 (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)))
65anbi1d 740 . . . . . 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 3299 . . . . 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 719 . . . 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 1065 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10 simpr2 1066 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11 colinearex 31862 . . . . . . . 8 Colinear ∈ V
1211cnvex 7075 . . . . . . 7 Colinear ∈ V
13 ecexg 7706 . . . . . . 7 ( Colinear ∈ V → [⟨𝐴, 𝐵⟩] Colinear ∈ V)
1412, 13ax-mp 5 . . . . . 6 [⟨𝐴, 𝐵⟩] Colinear ∈ V
15 eleq1 2686 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16 neeq1 2852 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝑏𝐴𝑏))
1715, 163anbi13d 1398 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏)))
18 opeq1 4377 . . . . . . . . . . 11 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
1918eceq1d 7743 . . . . . . . . . 10 (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩] Colinear = [⟨𝐴, 𝑏⟩] Colinear )
2019eqeq2d 2631 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ))
2117, 20anbi12d 746 . . . . . . . 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 3047 . . . . . . 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 2686 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24 neeq2 2853 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2523, 243anbi23d 1399 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵)))
26 opeq2 4378 . . . . . . . . . . 11 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
2726eceq1d 7743 . . . . . . . . . 10 (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )
2827eqeq2d 2631 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ))
2925, 28anbi12d 746 . . . . . . . 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 3047 . . . . . . 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 2625 . . . . . . . . 9 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (𝑙 = [⟨𝐴, 𝐵⟩] Colinear ↔ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
3231anbi2d 739 . . . . . . . 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 3047 . . . . . . 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 6713 . . . . . 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 1410 . . . . 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 692 . . . 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 6618 . . . 4 (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39 df-br 4624 . . . . . 6 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ Line)
40 df-line2 31939 . . . . . . 7 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
4140eleq2i 2690 . . . . . 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 31944 . . . . . 6 Fun Line
44 funbrfv 6201 . . . . . 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 2667 . . 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 4903 . . . 4 𝐴, 𝐵⟩ ∈ V
50 dfec2 7705 . . . 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 3193 . . . . 5 𝑥 ∈ V
5349, 52brcnv 5275 . . . 4 (⟨𝐴, 𝐵 Colinear 𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩)
5453abbii 2736 . . 3 {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5551, 54eqtri 2643 . 2 [⟨𝐴, 𝐵⟩] Colinear = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5648, 55syl6eq 2671 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 384  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wrex 2909  Vcvv 3190  cop 4161   class class class wbr 4623  ccnv 5083  Fun wfun 5851  cfv 5857  (class class class)co 6615  {coprab 6616  [cec 7700  cn 10980  𝔼cee 25702   Colinear ccolin 31839  Linecline2 31936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-i2m1 9964  ax-1ne0 9965  ax-rrecex 9968  ax-cnre 9969
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-ec 7704  df-nn 10981  df-colinear 31841  df-line2 31939
This theorem is referenced by:  liness  31947  fvline2  31948  ellines  31954
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