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Theorem fvline 35418
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 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ (𝐴Line𝐡) = {π‘₯ ∣ π‘₯ Colinear ⟨𝐴, 𝐡⟩})
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Allowed substitution hint:   𝑁(π‘₯)

Proof of Theorem fvline
Dummy variables π‘Ž 𝑏 𝑙 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . . . 5 [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear
2 fveq2 6892 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
32eleq2d 2817 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝐴 ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘)))
42eleq2d 2817 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝐡 ∈ (π”Όβ€˜π‘›) ↔ 𝐡 ∈ (π”Όβ€˜π‘)))
53, 43anbi12d 1435 . . . . . . 7 (𝑛 = 𝑁 β†’ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ↔ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)))
65anbi1d 628 . . . . . 6 (𝑛 = 𝑁 β†’ (((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
76rspcev 3613 . . . . 5 ((𝑁 ∈ β„• ∧ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )) β†’ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear ))
81, 7mpanr2 700 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear ))
9 simpr1 1192 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
10 simpr2 1193 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
11 colinearex 35334 . . . . . . . 8 Colinear ∈ V
1211cnvex 7920 . . . . . . 7 β—‘ Colinear ∈ V
13 ecexg 8711 . . . . . . 7 (β—‘ Colinear ∈ V β†’ [⟨𝐴, 𝐡⟩]β—‘ Colinear ∈ V)
1412, 13ax-mp 5 . . . . . 6 [⟨𝐴, 𝐡⟩]β—‘ Colinear ∈ V
15 eleq1 2819 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ (π‘Ž ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘›)))
16 neeq1 3001 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ (π‘Ž β‰  𝑏 ↔ 𝐴 β‰  𝑏))
1715, 163anbi13d 1436 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ↔ (𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏)))
18 opeq1 4874 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ βŸ¨π‘Ž, π‘βŸ© = ⟨𝐴, π‘βŸ©)
1918eceq1d 8746 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear = [⟨𝐴, π‘βŸ©]β—‘ Colinear )
2019eqeq2d 2741 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear ↔ 𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear ))
2117, 20anbi12d 629 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏) ∧ 𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear )))
2221rexbidv 3176 . . . . . . 7 (π‘Ž = 𝐴 β†’ (βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏) ∧ 𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear )))
23 eleq1 2819 . . . . . . . . . 10 (𝑏 = 𝐡 β†’ (𝑏 ∈ (π”Όβ€˜π‘›) ↔ 𝐡 ∈ (π”Όβ€˜π‘›)))
24 neeq2 3002 . . . . . . . . . 10 (𝑏 = 𝐡 β†’ (𝐴 β‰  𝑏 ↔ 𝐴 β‰  𝐡))
2523, 243anbi23d 1437 . . . . . . . . 9 (𝑏 = 𝐡 β†’ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏) ↔ (𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡)))
26 opeq2 4875 . . . . . . . . . . 11 (𝑏 = 𝐡 β†’ ⟨𝐴, π‘βŸ© = ⟨𝐴, 𝐡⟩)
2726eceq1d 8746 . . . . . . . . . 10 (𝑏 = 𝐡 β†’ [⟨𝐴, π‘βŸ©]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )
2827eqeq2d 2741 . . . . . . . . 9 (𝑏 = 𝐡 β†’ (𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear ↔ 𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear ))
2925, 28anbi12d 629 . . . . . . . 8 (𝑏 = 𝐡 β†’ (((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏) ∧ 𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ 𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
3029rexbidv 3176 . . . . . . 7 (𝑏 = 𝐡 β†’ (βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝑏) ∧ 𝑙 = [⟨𝐴, π‘βŸ©]β—‘ Colinear ) ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ 𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
31 eqeq1 2734 . . . . . . . . 9 (𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear β†’ (𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear ↔ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear ))
3231anbi2d 627 . . . . . . . 8 (𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear β†’ (((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ 𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear ) ↔ ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
3332rexbidv 3176 . . . . . . 7 (𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear β†’ (βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ 𝑙 = [⟨𝐴, 𝐡⟩]β—‘ Colinear ) ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
3422, 30, 33eloprabg 7522 . . . . . 6 ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear ∈ V) β†’ (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )} ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
3514, 34mp3an3 1448 . . . . 5 ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )} ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
369, 10, 35syl2anc 582 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )} ↔ βˆƒπ‘› ∈ β„• ((𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝐡 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 β‰  𝐡) ∧ [⟨𝐴, 𝐡⟩]β—‘ Colinear = [⟨𝐴, 𝐡⟩]β—‘ Colinear )))
378, 36mpbird 256 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ ⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )})
38 df-ov 7416 . . . 4 (𝐴Line𝐡) = (Lineβ€˜βŸ¨π΄, 𝐡⟩)
39 df-br 5150 . . . . . 6 (⟨𝐴, 𝐡⟩Line[⟨𝐴, 𝐡⟩]β—‘ Colinear ↔ ⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ Line)
40 df-line2 35411 . . . . . . 7 Line = {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )}
4140eleq2i 2823 . . . . . 6 (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ Line ↔ ⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )})
4239, 41bitri 274 . . . . 5 (⟨𝐴, 𝐡⟩Line[⟨𝐴, 𝐡⟩]β—‘ Colinear ↔ ⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )})
43 funline 35416 . . . . . 6 Fun Line
44 funbrfv 6943 . . . . . 6 (Fun Line β†’ (⟨𝐴, 𝐡⟩Line[⟨𝐴, 𝐡⟩]β—‘ Colinear β†’ (Lineβ€˜βŸ¨π΄, 𝐡⟩) = [⟨𝐴, 𝐡⟩]β—‘ Colinear ))
4543, 44ax-mp 5 . . . . 5 (⟨𝐴, 𝐡⟩Line[⟨𝐴, 𝐡⟩]β—‘ Colinear β†’ (Lineβ€˜βŸ¨π΄, 𝐡⟩) = [⟨𝐴, 𝐡⟩]β—‘ Colinear )
4642, 45sylbir 234 . . . 4 (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )} β†’ (Lineβ€˜βŸ¨π΄, 𝐡⟩) = [⟨𝐴, 𝐡⟩]β—‘ Colinear )
4738, 46eqtrid 2782 . . 3 (⟨⟨𝐴, 𝐡⟩, [⟨𝐴, 𝐡⟩]β—‘ Colinear ⟩ ∈ {βŸ¨βŸ¨π‘Ž, π‘βŸ©, π‘™βŸ© ∣ βˆƒπ‘› ∈ β„• ((π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑏 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž β‰  𝑏) ∧ 𝑙 = [βŸ¨π‘Ž, π‘βŸ©]β—‘ Colinear )} β†’ (𝐴Line𝐡) = [⟨𝐴, 𝐡⟩]β—‘ Colinear )
4837, 47syl 17 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ (𝐴Line𝐡) = [⟨𝐴, 𝐡⟩]β—‘ Colinear )
49 opex 5465 . . . 4 ⟨𝐴, 𝐡⟩ ∈ V
50 dfec2 8710 . . . 4 (⟨𝐴, 𝐡⟩ ∈ V β†’ [⟨𝐴, 𝐡⟩]β—‘ Colinear = {π‘₯ ∣ ⟨𝐴, π΅βŸ©β—‘ Colinear π‘₯})
5149, 50ax-mp 5 . . 3 [⟨𝐴, 𝐡⟩]β—‘ Colinear = {π‘₯ ∣ ⟨𝐴, π΅βŸ©β—‘ Colinear π‘₯}
52 vex 3476 . . . . 5 π‘₯ ∈ V
5349, 52brcnv 5883 . . . 4 (⟨𝐴, π΅βŸ©β—‘ Colinear π‘₯ ↔ π‘₯ Colinear ⟨𝐴, 𝐡⟩)
5453abbii 2800 . . 3 {π‘₯ ∣ ⟨𝐴, π΅βŸ©β—‘ Colinear π‘₯} = {π‘₯ ∣ π‘₯ Colinear ⟨𝐴, 𝐡⟩}
5551, 54eqtri 2758 . 2 [⟨𝐴, 𝐡⟩]β—‘ Colinear = {π‘₯ ∣ π‘₯ Colinear ⟨𝐴, 𝐡⟩}
5648, 55eqtrdi 2786 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 β‰  𝐡)) β†’ (𝐴Line𝐡) = {π‘₯ ∣ π‘₯ Colinear ⟨𝐴, 𝐡⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {cab 2707   β‰  wne 2938  βˆƒwrex 3068  Vcvv 3472  βŸ¨cop 4635   class class class wbr 5149  β—‘ccnv 5676  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7413  {coprab 7414  [cec 8705  β„•cn 12218  π”Όcee 28411   Colinear ccolin 35311  Linecline2 35408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-1cn 11172  ax-addcl 11174
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-ec 8709  df-nn 12219  df-colinear 35313  df-line2 35411
This theorem is referenced by:  liness  35419  fvline2  35420  ellines  35426
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