Step | Hyp | Ref
| Expression |
1 | | nnex 12164 |
. . . 4
β’ β
β V |
2 | 1, 1 | xpex 7688 |
. . 3
β’ (β
Γ β) β V |
3 | | opabssxp 5725 |
. . 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 5280 |
. 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 770 |
. . . . . . . 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 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 0 < π) |
7 | | qgt0numnn 16631 |
. . . . . . . 8
β’ ((π β β β§ 0 <
π) β
(numerβπ) β
β) |
8 | 5, 6, 7 | syl2anc 585 |
. . . . . . 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 16621 |
. . . . . . . 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 513 |
. . . . . 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 766 |
. . . . . . 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 768 |
. . . . . . 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 41195 |
. . . . . . 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 1374 |
. . . . . 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 781 |
. . . . . . . 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 16626 |
. . . . . . . . . . . 12
β’ (π β β β π = ((numerβπ) / (denomβπ))) |
18 | 17 | oveq1d 7373 |
. . . . . . . . . . 11
β’ (π β β β (π β (ββπ·)) = (((numerβπ) / (denomβπ)) β (ββπ·))) |
19 | 18 | fveq2d 6847 |
. . . . . . . . . 10
β’ (π β β β
(absβ(π β
(ββπ·))) =
(absβ(((numerβπ) / (denomβπ)) β (ββπ·)))) |
20 | 19 | breq1d 5116 |
. . . . . . . . 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 41196 |
. . . . . . 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 1374 |
. . . . . 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 517 |
. . . . 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 414 |
. . . 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 5110 |
. . . . . 6
β’ (π₯ = π β (0 < π₯ β 0 < π)) |
28 | | fvoveq1 7381 |
. . . . . . 7
β’ (π₯ = π β (absβ(π₯ β (ββπ·))) = (absβ(π β (ββπ·)))) |
29 | | fveq2 6843 |
. . . . . . . 8
β’ (π₯ = π β (denomβπ₯) = (denomβπ)) |
30 | 29 | oveq1d 7373 |
. . . . . . 7
β’ (π₯ = π β ((denomβπ₯)β-2) = ((denomβπ)β-2)) |
31 | 28, 30 | breq12d 5119 |
. . . . . 6
β’ (π₯ = π β ((absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2) β (absβ(π β (ββπ·))) < ((denomβπ)β-2))) |
32 | 27, 31 | anbi12d 632 |
. . . . 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 3646 |
. . . 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 6856 |
. . . . 5
β’
(numerβπ)
β V |
35 | | fvex 6856 |
. . . . 5
β’
(denomβπ)
β V |
36 | | eleq1 2822 |
. . . . . . 7
β’ (π¦ = (numerβπ) β (π¦ β β β (numerβπ) β
β)) |
37 | 36 | anbi1d 631 |
. . . . . 6
β’ (π¦ = (numerβπ) β ((π¦ β β β§ π§ β β) β ((numerβπ) β β β§ π§ β
β))) |
38 | | oveq1 7365 |
. . . . . . . . 9
β’ (π¦ = (numerβπ) β (π¦β2) = ((numerβπ)β2)) |
39 | 38 | oveq1d 7373 |
. . . . . . . 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 6847 |
. . . . . . . 8
β’ (π¦ = (numerβπ) β (absβ((π¦β2) β (π· Β· (π§β2)))) = (absβ(((numerβπ)β2) β (π· Β· (π§β2))))) |
42 | 41 | breq1d 5116 |
. . . . . . 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 632 |
. . . . . 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 632 |
. . . . 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 2822 |
. . . . . . 7
β’ (π§ = (denomβπ) β (π§ β β β (denomβπ) β
β)) |
46 | 45 | anbi2d 630 |
. . . . . 6
β’ (π§ = (denomβπ) β (((numerβπ) β β β§ π§ β β) β
((numerβπ) β
β β§ (denomβπ) β β))) |
47 | | oveq1 7365 |
. . . . . . . . . 10
β’ (π§ = (denomβπ) β (π§β2) = ((denomβπ)β2)) |
48 | 47 | oveq2d 7374 |
. . . . . . . . 9
β’ (π§ = (denomβπ) β (π· Β· (π§β2)) = (π· Β· ((denomβπ)β2))) |
49 | 48 | oveq2d 7374 |
. . . . . . . 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 6847 |
. . . . . . . 8
β’ (π§ = (denomβπ) β
(absβ(((numerβπ)β2) β (π· Β· (π§β2)))) = (absβ(((numerβπ)β2) β (π· Β· ((denomβπ)β2))))) |
52 | 51 | breq1d 5116 |
. . . . . . 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 632 |
. . . . . 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 632 |
. . . . 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 5500 |
. . . 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 296 |
. . 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 4038 |
. . . . . 6
β’ {π₯ β β β£ (0 <
π₯ β§ (absβ(π₯ β (ββπ·))) < ((denomβπ₯)β-2))} β
β |
58 | | simprl 770 |
. . . . . 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 3943 |
. . . . 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 772 |
. . . . . 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 3943 |
. . . . 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 5434 |
. . . . . . 7
β’
(β¨(numerβπ), (denomβπ)β© = β¨(numerβπ), (denomβπ)β© β
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) |
63 | | simprl 770 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
(numerβπ) =
(numerβπ)) |
64 | | simprr 772 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
(denomβπ) =
(denomβπ)) |
65 | 63, 64 | oveq12d 7376 |
. . . . . . . . 9
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
((numerβπ) /
(denomβπ)) =
((numerβπ) /
(denomβπ))) |
66 | | simpll 766 |
. . . . . . . . . 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 768 |
. . . . . . . . . 10
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π β
β) |
69 | | qeqnumdivden 16626 |
. . . . . . . . . 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 2783 |
. . . . . . . 8
β’ (((π β β β§ π β β) β§
((numerβπ) =
(numerβπ) β§
(denomβπ) =
(denomβπ))) β
π = π) |
72 | 71 | ex 414 |
. . . . . . 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 6843 |
. . . . . . 7
β’ (π = π β (numerβπ) = (numerβπ)) |
75 | | fveq2 6843 |
. . . . . . 7
β’ (π = π β (denomβπ) = (denomβπ)) |
76 | 74, 75 | opeq12d 4839 |
. . . . . 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 585 |
. . . 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 414 |
. . 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 8936 |
. 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·)))))}) |