| Step | Hyp | Ref
| Expression |
|---|
| 1 | | nnex 12220 |
. . . 4
β’ β
β V |
| 2 | 1, 1 | xpex 7742 |
. . 3
β’ (β
Γ β) β V |
| 3 | | opabssxp 5768 |
. . 3
β’
{β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))}
β (β Γ β) |
| 4 | 2, 3 | ssexi 5322 |
. 2
β’
{β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))}
β V |
| 5 | | simprl 769 |
. . . . . . . 8
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β π β β) |
| 6 | | simprrl 779 |
. . . . . . . 8
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β 0 < π) |
| 7 | | qgt0numnn 16689 |
. . . . . . . 8
β’ ((π β β β§ 0 <
π) β
(numerβπ) β
β) |
| 8 | 5, 6, 7 | syl2anc 584 |
. . . . . . 7
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β (numerβπ) β
β) |
| 9 | | qdencl 16679 |
. . . . . . . 8
β’ (π β β β
(denomβπ) β
β) |
| 10 | 5, 9 | syl 17 |
. . . . . . 7
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β (denomβπ) β
β) |
| 11 | 8, 10 | jca 512 |
. . . . . 6
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β ((numerβπ) β β β§
(denomβπ) β
β)) |
| 12 | | simpll 765 |
. . . . . . 7
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β π· β β) |
| 13 | | simplr 767 |
. . . . . . 7
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β Β¬
(ββπ·) β
β) |
| 14 | | pellexlem1 41649 |
. . . . . . 7
β’ (((π· β β β§
(numerβπ) β
β β§ (denomβπ) β β) β§ Β¬
(ββπ·) β
β) β (((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β 0) |
| 15 | 12, 8, 10, 13, 14 | syl31anc 1373 |
. . . . . 6
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β (((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β
0) |
| 16 | | simprrr 780 |
. . . . . . . 8
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β (absβ(π β (ββπ·))) < ((denomβπ)β-2)) |
| 17 | | qeqnumdivden 16684 |
. . . . . . . . . . . 12
β’ (π β β β π = ((numerβπ) / (denomβπ))) |
| 18 | 17 | oveq1d 7426 |
. . . . . . . . . . 11
β’ (π β β β (π β (ββπ·)) = (((numerβπ) / (denomβπ)) β (ββπ·))) |
| 19 | 18 | fveq2d 6895 |
. . . . . . . . . 10
β’ (π β β β
(absβ(π β
(ββπ·))) =
(absβ(((numerβπ) / (denomβπ)) β (ββπ·)))) |
| 20 | 19 | breq1d 5158 |
. . . . . . . . 9
β’ (π β β β
((absβ(π β
(ββπ·))) <
((denomβπ)β-2)
β (absβ(((numerβπ) / (denomβπ)) β (ββπ·))) < ((denomβπ)β-2))) |
| 21 | 5, 20 | syl 17 |
. . . . . . . 8
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β ((absβ(π β (ββπ·))) < ((denomβπ)β-2) β
(absβ(((numerβπ) / (denomβπ)) β (ββπ·))) < ((denomβπ)β-2))) |
| 22 | 16, 21 | mpbid 231 |
. . . . . . 7
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β
(absβ(((numerβπ) / (denomβπ)) β (ββπ·))) < ((denomβπ)β-2)) |
| 23 | | pellexlem2 41650 |
. . . . . . 7
β’ (((π· β β β§
(numerβπ) β
β β§ (denomβπ) β β) β§
(absβ(((numerβπ) / (denomβπ)) β (ββπ·))) < ((denomβπ)β-2)) β
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·)))) |
| 24 | 12, 8, 10, 22, 23 | syl31anc 1373 |
. . . . . 6
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·)))) |
| 25 | 11, 15, 24 | jca32 516 |
. . . . 5
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))) β (((numerβπ) β β β§
(denomβπ) β
β) β§ ((((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·)))))) |
| 26 | 25 | ex 413 |
. . . 4
β’ ((π· β β β§ Β¬
(ββπ·) β
β) β ((π β
β β§ (0 < π
β§ (absβ(π β
(ββπ·))) <
((denomβπ)β-2)))
β (((numerβπ)
β β β§ (denomβπ) β β) β§ ((((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·))))))) |
| 27 | | breq2 5152 |
. . . . . 6
β’ (π₯ = π β (0 < π₯ β 0 < π)) |
| 28 | | fvoveq1 7434 |
. . . . . . 7
β’ (π₯ = π β (absβ(π₯ β (ββπ·))) = (absβ(π β (ββπ·)))) |
| 29 | | fveq2 6891 |
. . . . . . . 8
β’ (π₯ = π β (denomβπ₯) = (denomβπ)) |
| 30 | 29 | oveq1d 7426 |
. . . . . . 7
β’ (π₯ = π β ((denomβπ₯)β-2) = ((denomβπ)β-2)) |
| 31 | 28, 30 | breq12d 5161 |
. . . . . 6
β’ (π₯ = π β ((absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2) β (absβ(π β (ββπ·))) < ((denomβπ)β-2))) |
| 32 | 27, 31 | anbi12d 631 |
. . . . 5
β’ (π₯ = π β ((0 < π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2)) β (0 < π β§ (absβ(π β (ββπ·))) < ((denomβπ)β-2)))) |
| 33 | 32 | elrab 3683 |
. . . 4
β’ (π β {π₯ β β β£ (0 < π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))} β (π β β β§ (0 <
π β§ (absβ(π β (ββπ·))) < ((denomβπ)β-2)))) |
| 34 | | fvex 6904 |
. . . . 5
β’
(numerβπ)
β V |
| 35 | | fvex 6904 |
. . . . 5
β’
(denomβπ)
β V |
| 36 | | eleq1 2821 |
. . . . . . 7
β’ (π¦ = (numerβπ) β (π¦ β β β (numerβπ) β
β)) |
| 37 | 36 | anbi1d 630 |
. . . . . 6
β’ (π¦ = (numerβπ) β ((π¦ β β β§ π§ β β) β ((numerβπ) β β β§ π§ β
β))) |
| 38 | | oveq1 7418 |
. . . . . . . . 9
β’ (π¦ = (numerβπ) β (π¦β2) = ((numerβπ)β2)) |
| 39 | 38 | oveq1d 7426 |
. . . . . . . 8
β’ (π¦ = (numerβπ) β ((π¦β2) β (π· Β· (π§β2))) = (((numerβπ)β2) β (π· Β· (π§β2)))) |
| 40 | 39 | neeq1d 3000 |
. . . . . . 7
β’ (π¦ = (numerβπ) β (((π¦β2) β (π· Β· (π§β2))) β 0 β (((numerβπ)β2) β (π· Β· (π§β2))) β 0)) |
| 41 | 39 | fveq2d 6895 |
. . . . . . . 8
β’ (π¦ = (numerβπ) β (absβ((π¦β2) β (π· Β· (π§β2)))) = (absβ(((numerβπ)β2) β (π· Β· (π§β2))))) |
| 42 | 41 | breq1d 5158 |
. . . . . . 7
β’ (π¦ = (numerβπ) β ((absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·))) β
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·))))) |
| 43 | 40, 42 | anbi12d 631 |
. . . . . 6
β’ (π¦ = (numerβπ) β ((((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))) β
((((numerβπ)β2)
β (π· Β· (π§β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))) |
| 44 | 37, 43 | anbi12d 631 |
. . . . 5
β’ (π¦ = (numerβπ) β (((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))
β (((numerβπ)
β β β§ π§
β β) β§ ((((numerβπ)β2) β (π· Β· (π§β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·))))))) |
| 45 | | eleq1 2821 |
. . . . . . 7
β’ (π§ = (denomβπ) β (π§ β β β (denomβπ) β
β)) |
| 46 | 45 | anbi2d 629 |
. . . . . 6
β’ (π§ = (denomβπ) β (((numerβπ) β β β§ π§ β β) β
((numerβπ) β
β β§ (denomβπ) β β))) |
| 47 | | oveq1 7418 |
. . . . . . . . . 10
β’ (π§ = (denomβπ) β (π§β2) = ((denomβπ)β2)) |
| 48 | 47 | oveq2d 7427 |
. . . . . . . . 9
β’ (π§ = (denomβπ) β (π· Β· (π§β2)) = (π· Β· ((denomβπ)β2))) |
| 49 | 48 | oveq2d 7427 |
. . . . . . . 8
β’ (π§ = (denomβπ) β (((numerβπ)β2) β (π· Β· (π§β2))) = (((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) |
| 50 | 49 | neeq1d 3000 |
. . . . . . 7
β’ (π§ = (denomβπ) β ((((numerβπ)β2) β (π· Β· (π§β2))) β 0 β (((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β
0)) |
| 51 | 49 | fveq2d 6895 |
. . . . . . . 8
β’ (π§ = (denomβπ) β
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) = (absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2))))) |
| 52 | 51 | breq1d 5158 |
. . . . . . 7
β’ (π§ = (denomβπ) β
((absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·))) β
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·))))) |
| 53 | 50, 52 | anbi12d 631 |
. . . . . 6
β’ (π§ = (denomβπ) β (((((numerβπ)β2) β (π· Β· (π§β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))) β
((((numerβπ)β2)
β (π· Β·
((denomβπ)β2)))
β 0 β§ (absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·)))))) |
| 54 | 46, 53 | anbi12d 631 |
. . . . 5
β’ (π§ = (denomβπ) β ((((numerβπ) β β β§ π§ β β) β§
((((numerβπ)β2)
β (π· Β· (π§β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))
β (((numerβπ)
β β β§ (denomβπ) β β) β§ ((((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·))))))) |
| 55 | 34, 35, 44, 54 | opelopab 5542 |
. . . 4
β’
(β¨(numerβπ), (denomβπ)β© β {β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))}
β (((numerβπ)
β β β§ (denomβπ) β β) β§ ((((numerβπ)β2) β (π· Β· ((denomβπ)β2))) β 0 β§
(absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2)))) < (1 + (2 Β·
(ββπ·)))))) |
| 56 | 26, 33, 55 | 3imtr4g 295 |
. . 3
β’ ((π· β β β§ Β¬
(ββπ·) β
β) β (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β β¨(numerβπ), (denomβπ)β© β {β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))})) |
| 57 | | ssrab2 4077 |
. . . . . 6
β’ {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))} β
β |
| 58 | | simprl 769 |
. . . . . 6
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))})) β π β {π₯ β β β£ (0 < π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))}) |
| 59 | 57, 58 | sselid 3980 |
. . . . 5
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))})) β π β
β) |
| 60 | | simprr 771 |
. . . . . 6
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))})) β π β {π₯ β β β£ (0 < π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))}) |
| 61 | 57, 60 | sselid 3980 |
. . . . 5
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))})) β π β
β) |
| 62 | 34, 35 | opth 5476 |
. . . . . . 7
β’
(β¨(numerβπ), (denomβπ)β© = β¨(numerβπ), (denomβπ)β© β
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) |
| 63 | | simprl 769 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
(numerβπ) =
(numerβπ)) |
| 64 | | simprr 771 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
(denomβπ) =
(denomβπ)) |
| 65 | 63, 64 | oveq12d 7429 |
. . . . . . . . 9
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
((numerβπ) /
(denomβπ)) =
((numerβπ) /
(denomβπ))) |
| 66 | | simpll 765 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π β
β) |
| 67 | 66, 17 | syl 17 |
. . . . . . . . 9
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π = ((numerβπ) / (denomβπ))) |
| 68 | | simplr 767 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π β
β) |
| 69 | | qeqnumdivden 16684 |
. . . . . . . . . 10
β’ (π β β β π = ((numerβπ) / (denomβπ))) |
| 70 | 68, 69 | syl 17 |
. . . . . . . . 9
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π = ((numerβπ) / (denomβπ))) |
| 71 | 65, 67, 70 | 3eqtr4d 2782 |
. . . . . . . 8
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π = π) |
| 72 | 71 | ex 413 |
. . . . . . 7
β’ ((π β β β§ π β β) β
(((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ)) β
π = π)) |
| 73 | 62, 72 | biimtrid 241 |
. . . . . 6
β’ ((π β β β§ π β β) β
(β¨(numerβπ),
(denomβπ)β© =
β¨(numerβπ),
(denomβπ)β©
β π = π)) |
| 74 | | fveq2 6891 |
. . . . . . 7
β’ (π = π β (numerβπ) = (numerβπ)) |
| 75 | | fveq2 6891 |
. . . . . . 7
β’ (π = π β (denomβπ) = (denomβπ)) |
| 76 | 74, 75 | opeq12d 4881 |
. . . . . 6
β’ (π = π β β¨(numerβπ), (denomβπ)β© = β¨(numerβπ), (denomβπ)β©) |
| 77 | 73, 76 | impbid1 224 |
. . . . 5
β’ ((π β β β§ π β β) β
(β¨(numerβπ),
(denomβπ)β© =
β¨(numerβπ),
(denomβπ)β©
β π = π)) |
| 78 | 59, 61, 77 | syl2anc 584 |
. . . 4
β’ (((π· β β β§ Β¬
(ββπ·) β
β) β§ (π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))})) β
(β¨(numerβπ),
(denomβπ)β© =
β¨(numerβπ),
(denomβπ)β©
β π = π)) |
| 79 | 78 | ex 413 |
. . 3
β’ ((π· β β β§ Β¬
(ββπ·) β
β) β ((π β
{π₯ β β β£
(0 < π₯ β§
(absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
β§ π β {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))}) β
(β¨(numerβπ),
(denomβπ)β© =
β¨(numerβπ),
(denomβπ)β©
β π = π))) |
| 80 | 56, 79 | dom2d 8991 |
. 2
β’ ((π· β β β§ Β¬
(ββπ·) β
β) β ({β¨π¦,
π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))}
β V β {π₯ β
β β£ (0 < π₯
β§ (absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
βΌ {β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))})) |
| 81 | 4, 80 | mpi 20 |
1
β’ ((π· β β β§ Β¬
(ββπ·) β
β) β {π₯ β
β β£ (0 < π₯
β§ (absβ(π₯ β
(ββπ·))) <
((denomβπ₯)β-2))}
βΌ {β¨π¦, π§β© β£ ((π¦ β β β§ π§ β β) β§ (((π¦β2) β (π· Β· (π§β2))) β 0 β§ (absβ((π¦β2) β (π· Β· (π§β2)))) < (1 + (2 Β·
(ββπ·)))))}) |
