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Theorem fvline 32570
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 2806 . . . . 5 [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear
2 fveq2 6404 . . . . . . . . 9 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
32eleq2d 2871 . . . . . . . 8 (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
42eleq2d 2871 . . . . . . . 8 (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
53, 43anbi12d 1554 . . . . . . 7 (𝑛 = 𝑁 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)))
65anbi1d 617 . . . . . 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 3502 . . . . 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 687 . . . 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 1241 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐴 ∈ (𝔼‘𝑁))
10 simpr2 1243 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → 𝐵 ∈ (𝔼‘𝑁))
11 colinearex 32486 . . . . . . . 8 Colinear ∈ V
1211cnvex 7339 . . . . . . 7 Colinear ∈ V
13 ecexg 7979 . . . . . . 7 ( Colinear ∈ V → [⟨𝐴, 𝐵⟩] Colinear ∈ V)
1412, 13ax-mp 5 . . . . . 6 [⟨𝐴, 𝐵⟩] Colinear ∈ V
15 eleq1 2873 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
16 neeq1 3040 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎𝑏𝐴𝑏))
1715, 163anbi13d 1555 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏)))
18 opeq1 4595 . . . . . . . . . . 11 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
1918eceq1d 8014 . . . . . . . . . 10 (𝑎 = 𝐴 → [⟨𝑎, 𝑏⟩] Colinear = [⟨𝐴, 𝑏⟩] Colinear )
2019eqeq2d 2816 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑙 = [⟨𝑎, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝑏⟩] Colinear ))
2117, 20anbi12d 618 . . . . . . . 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 3240 . . . . . . 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 2873 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
24 neeq2 3041 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2523, 243anbi23d 1556 . . . . . . . . 9 (𝑏 = 𝐵 → ((𝐴 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝐴𝑏) ↔ (𝐴 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑛) ∧ 𝐴𝐵)))
26 opeq2 4596 . . . . . . . . . . 11 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
2726eceq1d 8014 . . . . . . . . . 10 (𝑏 = 𝐵 → [⟨𝐴, 𝑏⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear )
2827eqeq2d 2816 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑙 = [⟨𝐴, 𝑏⟩] Colinear ↔ 𝑙 = [⟨𝐴, 𝐵⟩] Colinear ))
2925, 28anbi12d 618 . . . . . . . 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 3240 . . . . . . 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 2810 . . . . . . . . 9 (𝑙 = [⟨𝐴, 𝐵⟩] Colinear → (𝑙 = [⟨𝐴, 𝐵⟩] Colinear ↔ [⟨𝐴, 𝐵⟩] Colinear = [⟨𝐴, 𝐵⟩] Colinear ))
3231anbi2d 616 . . . . . . . 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 3240 . . . . . . 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 6974 . . . . . 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 1567 . . . . 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 575 . . . 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 248 . . 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 6873 . . . 4 (𝐴Line𝐵) = (Line‘⟨𝐴, 𝐵⟩)
39 df-br 4845 . . . . . 6 (⟨𝐴, 𝐵⟩Line[⟨𝐴, 𝐵⟩] Colinear ↔ ⟨⟨𝐴, 𝐵⟩, [⟨𝐴, 𝐵⟩] Colinear ⟩ ∈ Line)
40 df-line2 32563 . . . . . . 7 Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
4140eleq2i 2877 . . . . . 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 266 . . . . 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 32568 . . . . . 6 Fun Line
44 funbrfv 6450 . . . . . 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 226 . . . 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 2852 . . 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 5122 . . . 4 𝐴, 𝐵⟩ ∈ V
50 dfec2 7978 . . . 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 3394 . . . . 5 𝑥 ∈ V
5349, 52brcnv 5506 . . . 4 (⟨𝐴, 𝐵 Colinear 𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩)
5453abbii 2923 . . 3 {𝑥 ∣ ⟨𝐴, 𝐵 Colinear 𝑥} = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5551, 54eqtri 2828 . 2 [⟨𝐴, 𝐵⟩] Colinear = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩}
5648, 55syl6eq 2856 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wrex 3097  Vcvv 3391  cop 4376   class class class wbr 4844  ccnv 5310  Fun wfun 6091  cfv 6097  (class class class)co 6870  {coprab 6871  [cec 7973  cn 11301  𝔼cee 25981   Colinear ccolin 32463  Linecline2 32560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-i2m1 10285  ax-1ne0 10286  ax-rrecex 10289  ax-cnre 10290
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-ov 6873  df-oprab 6874  df-om 7292  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-ec 7977  df-nn 11302  df-colinear 32465  df-line2 32563
This theorem is referenced by:  liness  32571  fvline2  32572  ellines  32578
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