Step | Hyp | Ref
| Expression |
1 | | retop 24270 |
. . . 4
β’
(topGenβran (,)) β Top |
2 | | 0cld 22534 |
. . . 4
β’
((topGenβran (,)) β Top β β
β
(Clsdβ(topGenβran (,)))) |
3 | 1, 2 | ax-mp 5 |
. . 3
β’ β
β (Clsdβ(topGenβran (,))) |
4 | | simpl3 1194 |
. . . . 5
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ = β
) β π < (vol*βπ΄)) |
5 | | fveq2 6889 |
. . . . . 6
β’ (π΄ = β
β
(vol*βπ΄) =
(vol*ββ
)) |
6 | 5 | adantl 483 |
. . . . 5
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ = β
) β
(vol*βπ΄) =
(vol*ββ
)) |
7 | 4, 6 | breqtrd 5174 |
. . . 4
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ = β
) β π <
(vol*ββ
)) |
8 | | 0ss 4396 |
. . . 4
β’ β
β π΄ |
9 | 7, 8 | jctil 521 |
. . 3
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ = β
) β (β
β π΄ β§ π <
(vol*ββ
))) |
10 | | sseq1 4007 |
. . . . 5
β’ (π = β
β (π β π΄ β β
β π΄)) |
11 | | fveq2 6889 |
. . . . . 6
β’ (π = β
β
(vol*βπ ) =
(vol*ββ
)) |
12 | 11 | breq2d 5160 |
. . . . 5
β’ (π = β
β (π < (vol*βπ ) β π <
(vol*ββ
))) |
13 | 10, 12 | anbi12d 632 |
. . . 4
β’ (π = β
β ((π β π΄ β§ π < (vol*βπ )) β (β
β π΄ β§ π <
(vol*ββ
)))) |
14 | 13 | rspcev 3613 |
. . 3
β’ ((β
β (Clsdβ(topGenβran (,))) β§ (β
β π΄ β§ π < (vol*ββ
))) β
βπ β
(Clsdβ(topGenβran (,)))(π β π΄ β§ π < (vol*βπ ))) |
15 | 3, 9, 14 | sylancr 588 |
. 2
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ = β
) β βπ β
(Clsdβ(topGenβran (,)))(π β π΄ β§ π < (vol*βπ ))) |
16 | | mblfinlem1 36514 |
. . . 4
β’ ((π΄ β (topGenβran (,))
β§ π΄ β β
)
β βπ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
17 | 16 | 3ad2antl1 1186 |
. . 3
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ β β
) β
βπ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
18 | | simpl3 1194 |
. . . . . . . . 9
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β π < (vol*βπ΄)) |
19 | | f1ofo 6838 |
. . . . . . . . . . . . . . 15
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π:ββontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
20 | | rnco2 6250 |
. . . . . . . . . . . . . . . . 17
β’ ran ([,]
β π) = ([,] β
ran π) |
21 | | forn 6806 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran π = {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
22 | 21 | imaeq2d 6058 |
. . . . . . . . . . . . . . . . 17
β’ (π:ββontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ([,] β ran π) = ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
23 | 20, 22 | eqtrid 2785 |
. . . . . . . . . . . . . . . 16
β’ (π:ββontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran ([,] β π) = ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
24 | 23 | unieqd 4922 |
. . . . . . . . . . . . . . 15
β’ (π:ββontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ ran
([,] β π) = βͺ ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
25 | 19, 24 | syl 17 |
. . . . . . . . . . . . . 14
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ ran
([,] β π) = βͺ ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
26 | 25 | adantl 483 |
. . . . . . . . . . . . 13
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βͺ ran
([,] β π) = βͺ ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
27 | | oveq1 7413 |
. . . . . . . . . . . . . . . . 17
β’ (π₯ = π’ β (π₯ / (2βπ¦)) = (π’ / (2βπ¦))) |
28 | | oveq1 7413 |
. . . . . . . . . . . . . . . . . 18
β’ (π₯ = π’ β (π₯ + 1) = (π’ + 1)) |
29 | 28 | oveq1d 7421 |
. . . . . . . . . . . . . . . . 17
β’ (π₯ = π’ β ((π₯ + 1) / (2βπ¦)) = ((π’ + 1) / (2βπ¦))) |
30 | 27, 29 | opeq12d 4881 |
. . . . . . . . . . . . . . . 16
β’ (π₯ = π’ β β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β© = β¨(π’ / (2βπ¦)), ((π’ + 1) / (2βπ¦))β©) |
31 | | oveq2 7414 |
. . . . . . . . . . . . . . . . . 18
β’ (π¦ = π£ β (2βπ¦) = (2βπ£)) |
32 | 31 | oveq2d 7422 |
. . . . . . . . . . . . . . . . 17
β’ (π¦ = π£ β (π’ / (2βπ¦)) = (π’ / (2βπ£))) |
33 | 31 | oveq2d 7422 |
. . . . . . . . . . . . . . . . 17
β’ (π¦ = π£ β ((π’ + 1) / (2βπ¦)) = ((π’ + 1) / (2βπ£))) |
34 | 32, 33 | opeq12d 4881 |
. . . . . . . . . . . . . . . 16
β’ (π¦ = π£ β β¨(π’ / (2βπ¦)), ((π’ + 1) / (2βπ¦))β© = β¨(π’ / (2βπ£)), ((π’ + 1) / (2βπ£))β©) |
35 | 30, 34 | cbvmpov 7501 |
. . . . . . . . . . . . . . 15
β’ (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) = (π’ β β€, π£ β β0 β¦
β¨(π’ / (2βπ£)), ((π’ + 1) / (2βπ£))β©) |
36 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = π§ β ([,]βπ) = ([,]βπ§)) |
37 | 36 | sseq1d 4013 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π§ β (([,]βπ) β ([,]βπ) β ([,]βπ§) β ([,]βπ))) |
38 | | eqeq1 2737 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π§ β (π = π β π§ = π)) |
39 | 37, 38 | imbi12d 345 |
. . . . . . . . . . . . . . . . 17
β’ (π = π§ β ((([,]βπ) β ([,]βπ) β π = π) β (([,]βπ§) β ([,]βπ) β π§ = π))) |
40 | 39 | ralbidv 3178 |
. . . . . . . . . . . . . . . 16
β’ (π = π§ β (βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π) β βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ§) β ([,]βπ) β π§ = π))) |
41 | 40 | cbvrabv 3443 |
. . . . . . . . . . . . . . 15
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} = {π§ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ§) β ([,]βπ) β π§ = π)} |
42 | | ssrab2 4077 |
. . . . . . . . . . . . . . . 16
β’ {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) |
43 | 42 | a1i 11 |
. . . . . . . . . . . . . . 15
β’ ((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β
{π β ran (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£
([,]βπ) β π΄} β ran (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) |
44 | 35, 41, 43 | dyadmbllem 25108 |
. . . . . . . . . . . . . 14
β’ ((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β
βͺ ([,] β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄}) = βͺ ([,]
β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
45 | 44 | adantr 482 |
. . . . . . . . . . . . 13
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βͺ ([,]
β {π β ran
(π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£
([,]βπ) β π΄}) = βͺ ([,] β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)})) |
46 | 26, 45 | eqtr4d 2776 |
. . . . . . . . . . . 12
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βͺ ran
([,] β π) = βͺ ([,] β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄})) |
47 | | opnmbllem0 36513 |
. . . . . . . . . . . . . 14
β’ (π΄ β (topGenβran (,))
β βͺ ([,] β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄}) = π΄) |
48 | 47 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
β’ ((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β
βͺ ([,] β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄}) = π΄) |
49 | 48 | adantr 482 |
. . . . . . . . . . . 12
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βͺ ([,]
β {π β ran
(π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£
([,]βπ) β π΄}) = π΄) |
50 | 46, 49 | eqtrd 2773 |
. . . . . . . . . . 11
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βͺ ran
([,] β π) = π΄) |
51 | 50 | fveq2d 6893 |
. . . . . . . . . 10
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (vol*ββͺ ran ([,] β π)) = (vol*βπ΄)) |
52 | | f1of 6831 |
. . . . . . . . . . . . 13
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
53 | | ssrab2 4077 |
. . . . . . . . . . . . . 14
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} |
54 | 35 | dyadf 25100 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©):(β€ Γ
β0)βΆ( β€ β© (β Γ
β)) |
55 | | frn 6722 |
. . . . . . . . . . . . . . . 16
β’ ((π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©):(β€ Γ
β0)βΆ( β€ β© (β Γ β)) β ran
(π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β ( β€ β©
(β Γ β))) |
56 | 54, 55 | ax-mp 5 |
. . . . . . . . . . . . . . 15
β’ ran
(π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β ( β€ β©
(β Γ β)) |
57 | 42, 56 | sstri 3991 |
. . . . . . . . . . . . . 14
β’ {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β ( β€ β© (β Γ
β)) |
58 | 53, 57 | sstri 3991 |
. . . . . . . . . . . . 13
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ( β€ β© (β Γ
β)) |
59 | | fss 6732 |
. . . . . . . . . . . . 13
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ( β€ β© (β Γ
β))) β π:ββΆ( β€ β© (β Γ
β))) |
60 | 52, 58, 59 | sylancl 587 |
. . . . . . . . . . . 12
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π:ββΆ( β€ β© (β Γ
β))) |
61 | 53, 42 | sstri 3991 |
. . . . . . . . . . . . . . . . . . . 20
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) |
62 | | ffvelcdm 7081 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
63 | 61, 62 | sselid 3980 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (πβπ) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) |
64 | 63 | adantrr 716 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (πβπ) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) |
65 | | ffvelcdm 7081 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
66 | 61, 65 | sselid 3980 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) |
67 | 66 | adantrl 715 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (πβπ§) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) |
68 | 35 | dyaddisj 25105 |
. . . . . . . . . . . . . . . . . 18
β’ (((πβπ) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β§ (πβπ§) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©)) β (([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ)) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
69 | 64, 67, 68 | syl2anc 585 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ)) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
70 | 52, 69 | sylan 581 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ)) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
71 | | df-3or 1089 |
. . . . . . . . . . . . . . . 16
β’
((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ)) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
72 | 70, 71 | sylib 217 |
. . . . . . . . . . . . . . 15
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
73 | | elrabi 3677 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (πβπ§) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄}) |
74 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π = (πβπ) β ([,]βπ) = ([,]β(πβπ))) |
75 | 74 | sseq1d 4013 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π = (πβπ) β (([,]βπ) β ([,]βπ) β ([,]β(πβπ)) β ([,]βπ))) |
76 | | eqeq1 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π = (πβπ) β (π = π β (πβπ) = π)) |
77 | 75, 76 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π = (πβπ) β ((([,]βπ) β ([,]βπ) β π = π) β (([,]β(πβπ)) β ([,]βπ) β (πβπ) = π))) |
78 | 77 | ralbidv 3178 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π = (πβπ) β (βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π) β βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ)) β ([,]βπ) β (πβπ) = π))) |
79 | 78 | elrab 3683 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ((πβπ) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β§ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ)) β ([,]βπ) β (πβπ) = π))) |
80 | 79 | simprbi 498 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ)) β ([,]βπ) β (πβπ) = π)) |
81 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π = (πβπ§) β ([,]βπ) = ([,]β(πβπ§))) |
82 | 81 | sseq2d 4014 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π = (πβπ§) β (([,]β(πβπ)) β ([,]βπ) β ([,]β(πβπ)) β ([,]β(πβπ§)))) |
83 | | eqeq2 2745 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π = (πβπ§) β ((πβπ) = π β (πβπ) = (πβπ§))) |
84 | 82, 83 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π = (πβπ§) β ((([,]β(πβπ)) β ([,]βπ) β (πβπ) = π) β (([,]β(πβπ)) β ([,]β(πβπ§)) β (πβπ) = (πβπ§)))) |
85 | 84 | rspcva 3611 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((πβπ§) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β§ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ)) β ([,]βπ) β (πβπ) = π)) β (([,]β(πβπ)) β ([,]β(πβπ§)) β (πβπ) = (πβπ§))) |
86 | 73, 80, 85 | syl2anr 598 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (([,]β(πβπ)) β ([,]β(πβπ§)) β (πβπ) = (πβπ§))) |
87 | | elrabi 3677 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (πβπ) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄}) |
88 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π = (πβπ§) β ([,]βπ) = ([,]β(πβπ§))) |
89 | 88 | sseq1d 4013 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π = (πβπ§) β (([,]βπ) β ([,]βπ) β ([,]β(πβπ§)) β ([,]βπ))) |
90 | | eqeq1 2737 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π = (πβπ§) β (π = π β (πβπ§) = π)) |
91 | 89, 90 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π = (πβπ§) β ((([,]βπ) β ([,]βπ) β π = π) β (([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π))) |
92 | 91 | ralbidv 3178 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π = (πβπ§) β (βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π) β βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π))) |
93 | 92 | elrab 3683 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ((πβπ§) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β§ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π))) |
94 | 93 | simprbi 498 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π)) |
95 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π = (πβπ) β ([,]βπ) = ([,]β(πβπ))) |
96 | 95 | sseq2d 4014 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π = (πβπ) β (([,]β(πβπ§)) β ([,]βπ) β ([,]β(πβπ§)) β ([,]β(πβπ)))) |
97 | | eqeq2 2745 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π = (πβπ) β ((πβπ§) = π β (πβπ§) = (πβπ))) |
98 | 96, 97 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π = (πβπ) β ((([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π) β (([,]β(πβπ§)) β ([,]β(πβπ)) β (πβπ§) = (πβπ)))) |
99 | 98 | rspcva 3611 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((πβπ) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β§ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]β(πβπ§)) β ([,]βπ) β (πβπ§) = π)) β (([,]β(πβπ§)) β ([,]β(πβπ)) β (πβπ§) = (πβπ))) |
100 | 87, 94, 99 | syl2an 597 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (([,]β(πβπ§)) β ([,]β(πβπ)) β (πβπ§) = (πβπ))) |
101 | | eqcom 2740 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((πβπ§) = (πβπ) β (πβπ) = (πβπ§)) |
102 | 100, 101 | syl6ib 251 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (([,]β(πβπ§)) β ([,]β(πβπ)) β (πβπ) = (πβπ§))) |
103 | 86, 102 | jaod 858 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((πβπ) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (πβπ§) β {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β (πβπ) = (πβπ§))) |
104 | 62, 65, 103 | syl2an 597 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β)) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β (πβπ) = (πβπ§))) |
105 | 104 | anandis 677 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β (πβπ) = (πβπ§))) |
106 | 52, 105 | sylan 581 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β (πβπ) = (πβπ§))) |
107 | | f1of1 6830 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π:ββ1-1β{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
108 | | f1veqaeq 7253 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββ1-1β{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((πβπ) = (πβπ§) β π = π§)) |
109 | 107, 108 | sylan 581 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((πβπ) = (πβπ§) β π = π§)) |
110 | 106, 109 | syld 47 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β ((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β π = π§)) |
111 | 110 | orim1d 965 |
. . . . . . . . . . . . . . 15
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (((([,]β(πβπ)) β ([,]β(πβπ§)) β¨ ([,]β(πβπ§)) β ([,]β(πβπ))) β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
))) |
112 | 72, 111 | mpd 15 |
. . . . . . . . . . . . . 14
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
113 | 112 | ralrimivva 3201 |
. . . . . . . . . . . . 13
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ β β βπ§ β β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
114 | | eqeq1 2737 |
. . . . . . . . . . . . . . . . 17
β’ (π = π§ β (π = π β π§ = π)) |
115 | | 2fveq3 6894 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = π§ β ((,)β(πβπ)) = ((,)β(πβπ§))) |
116 | 115 | ineq1d 4211 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π§ β (((,)β(πβπ)) β© ((,)β(πβπ))) = (((,)β(πβπ§)) β© ((,)β(πβπ)))) |
117 | 116 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . 17
β’ (π = π§ β ((((,)β(πβπ)) β© ((,)β(πβπ))) = β
β (((,)β(πβπ§)) β© ((,)β(πβπ))) = β
)) |
118 | 114, 117 | orbi12d 918 |
. . . . . . . . . . . . . . . 16
β’ (π = π§ β ((π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
) β (π§ = π β¨ (((,)β(πβπ§)) β© ((,)β(πβπ))) = β
))) |
119 | 118 | ralbidv 3178 |
. . . . . . . . . . . . . . 15
β’ (π = π§ β (βπ β β (π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
) β βπ β β (π§ = π β¨ (((,)β(πβπ§)) β© ((,)β(πβπ))) = β
))) |
120 | 119 | cbvralvw 3235 |
. . . . . . . . . . . . . 14
β’
(βπ β
β βπ β
β (π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
) β βπ§ β β βπ β β (π§ = π β¨ (((,)β(πβπ§)) β© ((,)β(πβπ))) = β
)) |
121 | | eqeq2 2745 |
. . . . . . . . . . . . . . . . 17
β’ (π§ = π β (π = π§ β π = π)) |
122 | | 2fveq3 6894 |
. . . . . . . . . . . . . . . . . . 19
β’ (π§ = π β ((,)β(πβπ§)) = ((,)β(πβπ))) |
123 | 122 | ineq2d 4212 |
. . . . . . . . . . . . . . . . . 18
β’ (π§ = π β (((,)β(πβπ)) β© ((,)β(πβπ§))) = (((,)β(πβπ)) β© ((,)β(πβπ)))) |
124 | 123 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . 17
β’ (π§ = π β ((((,)β(πβπ)) β© ((,)β(πβπ§))) = β
β (((,)β(πβπ)) β© ((,)β(πβπ))) = β
)) |
125 | 121, 124 | orbi12d 918 |
. . . . . . . . . . . . . . . 16
β’ (π§ = π β ((π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β (π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
))) |
126 | 125 | cbvralvw 3235 |
. . . . . . . . . . . . . . 15
β’
(βπ§ β
β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β βπ β β (π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
)) |
127 | 126 | ralbii 3094 |
. . . . . . . . . . . . . 14
β’
(βπ β
β βπ§ β
β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β βπ β β βπ β β (π = π β¨ (((,)β(πβπ)) β© ((,)β(πβπ))) = β
)) |
128 | 122 | disjor 5128 |
. . . . . . . . . . . . . 14
β’
(Disj π§
β β ((,)β(πβπ§)) β βπ§ β β βπ β β (π§ = π β¨ (((,)β(πβπ§)) β© ((,)β(πβπ))) = β
)) |
129 | 120, 127,
128 | 3bitr4ri 304 |
. . . . . . . . . . . . 13
β’
(Disj π§
β β ((,)β(πβπ§)) β βπ β β βπ§ β β (π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
)) |
130 | 113, 129 | sylibr 233 |
. . . . . . . . . . . 12
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β Disj π§ β β ((,)β(πβπ§))) |
131 | | eqid 2733 |
. . . . . . . . . . . 12
β’ seq1( + ,
((abs β β ) β π)) = seq1( + , ((abs β β )
β π)) |
132 | 60, 130, 131 | uniiccvol 25089 |
. . . . . . . . . . 11
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (vol*ββͺ ran ([,] β π)) = sup(ran seq1( + , ((abs β β
) β π)),
β*, < )) |
133 | 132 | adantl 483 |
. . . . . . . . . 10
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (vol*ββͺ ran ([,] β π)) = sup(ran seq1( + , ((abs β β
) β π)),
β*, < )) |
134 | 51, 133 | eqtr3d 2775 |
. . . . . . . . 9
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (vol*βπ΄) = sup(ran seq1( + , ((abs β β
) β π)),
β*, < )) |
135 | 18, 134 | breqtrd 5174 |
. . . . . . . 8
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β π < sup(ran seq1( + , ((abs β
β ) β π)),
β*, < )) |
136 | | absf 15281 |
. . . . . . . . . . . 12
β’
abs:ββΆβ |
137 | | subf 11459 |
. . . . . . . . . . . 12
β’ β
:(β Γ β)βΆβ |
138 | | fco 6739 |
. . . . . . . . . . . 12
β’
((abs:ββΆβ β§ β :(β Γ
β)βΆβ) β (abs β β ):(β Γ
β)βΆβ) |
139 | 136, 137,
138 | mp2an 691 |
. . . . . . . . . . 11
β’ (abs
β β ):(β Γ β)βΆβ |
140 | | zre 12559 |
. . . . . . . . . . . . . . . . . . 19
β’ (π₯ β β€ β π₯ β
β) |
141 | | 2re 12283 |
. . . . . . . . . . . . . . . . . . . . 21
β’ 2 β
β |
142 | | reexpcl 14041 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((2
β β β§ π¦
β β0) β (2βπ¦) β β) |
143 | 141, 142 | mpan 689 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π¦ β β0
β (2βπ¦) β
β) |
144 | | 2cn 12284 |
. . . . . . . . . . . . . . . . . . . . 21
β’ 2 β
β |
145 | | 2ne0 12313 |
. . . . . . . . . . . . . . . . . . . . 21
β’ 2 β
0 |
146 | | nn0z 12580 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π¦ β β0
β π¦ β
β€) |
147 | | expne0i 14057 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((2
β β β§ 2 β 0 β§ π¦ β β€) β (2βπ¦) β 0) |
148 | 144, 145,
146, 147 | mp3an12i 1466 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π¦ β β0
β (2βπ¦) β
0) |
149 | 143, 148 | jca 513 |
. . . . . . . . . . . . . . . . . . 19
β’ (π¦ β β0
β ((2βπ¦) β
β β§ (2βπ¦)
β 0)) |
150 | | redivcl 11930 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β β β§
(2βπ¦) β β
β§ (2βπ¦) β 0)
β (π₯ / (2βπ¦)) β
β) |
151 | | peano2re 11384 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π₯ β β β (π₯ + 1) β
β) |
152 | | redivcl 11930 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π₯ + 1) β β β§
(2βπ¦) β β
β§ (2βπ¦) β 0)
β ((π₯ + 1) /
(2βπ¦)) β
β) |
153 | 151, 152 | syl3an1 1164 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π₯ β β β§
(2βπ¦) β β
β§ (2βπ¦) β 0)
β ((π₯ + 1) /
(2βπ¦)) β
β) |
154 | 150, 153 | opelxpd 5714 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π₯ β β β§
(2βπ¦) β β
β§ (2βπ¦) β 0)
β β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β© β (β Γ
β)) |
155 | 154 | 3expb 1121 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π₯ β β β§
((2βπ¦) β β
β§ (2βπ¦) β 0))
β β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β© β (β Γ
β)) |
156 | 140, 149,
155 | syl2an 597 |
. . . . . . . . . . . . . . . . . 18
β’ ((π₯ β β€ β§ π¦ β β0)
β β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β© β (β Γ
β)) |
157 | 156 | rgen2 3198 |
. . . . . . . . . . . . . . . . 17
β’
βπ₯ β
β€ βπ¦ β
β0 β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β© β (β Γ
β) |
158 | | eqid 2733 |
. . . . . . . . . . . . . . . . . 18
β’ (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) = (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) |
159 | 158 | fmpo 8051 |
. . . . . . . . . . . . . . . . 17
β’
(βπ₯ β
β€ βπ¦ β
β0 β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β© β (β Γ β)
β (π₯ β β€,
π¦ β
β0 β¦ β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©):(β€ Γ
β0)βΆ(β Γ β)) |
160 | 157, 159 | mpbi 229 |
. . . . . . . . . . . . . . . 16
β’ (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©):(β€ Γ
β0)βΆ(β Γ β) |
161 | | frn 6722 |
. . . . . . . . . . . . . . . 16
β’ ((π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©):(β€ Γ
β0)βΆ(β Γ β) β ran (π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β (β
Γ β)) |
162 | 160, 161 | ax-mp 5 |
. . . . . . . . . . . . . . 15
β’ ran
(π₯ β β€, π¦ β β0
β¦ β¨(π₯ /
(2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β (β
Γ β) |
163 | 42, 162 | sstri 3991 |
. . . . . . . . . . . . . 14
β’ {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β (β Γ
β) |
164 | 53, 163 | sstri 3991 |
. . . . . . . . . . . . 13
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (β Γ
β) |
165 | | ax-resscn 11164 |
. . . . . . . . . . . . . 14
β’ β
β β |
166 | | xpss12 5691 |
. . . . . . . . . . . . . 14
β’ ((β
β β β§ β β β) β (β Γ β)
β (β Γ β)) |
167 | 165, 165,
166 | mp2an 691 |
. . . . . . . . . . . . 13
β’ (β
Γ β) β (β Γ β) |
168 | 164, 167 | sstri 3991 |
. . . . . . . . . . . 12
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (β Γ
β) |
169 | | fss 6732 |
. . . . . . . . . . . 12
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (β Γ β)) β
π:ββΆ(β
Γ β)) |
170 | 168, 169 | mpan2 690 |
. . . . . . . . . . 11
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π:ββΆ(β Γ
β)) |
171 | | fco 6739 |
. . . . . . . . . . 11
β’ (((abs
β β ):(β Γ β)βΆβ β§ π:ββΆ(β Γ
β)) β ((abs β β ) β π):ββΆβ) |
172 | 139, 170,
171 | sylancr 588 |
. . . . . . . . . 10
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ((abs β β ) β
π):ββΆβ) |
173 | | nnuz 12862 |
. . . . . . . . . . 11
β’ β =
(β€β₯β1) |
174 | | 1z 12589 |
. . . . . . . . . . . 12
β’ 1 β
β€ |
175 | 174 | a1i 11 |
. . . . . . . . . . 11
β’ (((abs
β β ) β π):ββΆβ β 1 β
β€) |
176 | | ffvelcdm 7081 |
. . . . . . . . . . 11
β’ ((((abs
β β ) β π):ββΆβ β§ π β β) β (((abs
β β ) β π)βπ) β β) |
177 | 173, 175,
176 | serfre 13994 |
. . . . . . . . . 10
β’ (((abs
β β ) β π):ββΆβ β seq1( + ,
((abs β β ) β π)):ββΆβ) |
178 | | frn 6722 |
. . . . . . . . . . 11
β’ (seq1( +
, ((abs β β ) β π)):ββΆβ β ran seq1( +
, ((abs β β ) β π)) β β) |
179 | | ressxr 11255 |
. . . . . . . . . . 11
β’ β
β β* |
180 | 178, 179 | sstrdi 3994 |
. . . . . . . . . 10
β’ (seq1( +
, ((abs β β ) β π)):ββΆβ β ran seq1( +
, ((abs β β ) β π)) β
β*) |
181 | 52, 172, 177, 180 | 4syl 19 |
. . . . . . . . 9
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran seq1( + , ((abs β β
) β π)) β
β*) |
182 | | rexr 11257 |
. . . . . . . . . 10
β’ (π β β β π β
β*) |
183 | 182 | 3ad2ant2 1135 |
. . . . . . . . 9
β’ ((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β
π β
β*) |
184 | | supxrlub 13301 |
. . . . . . . . 9
β’ ((ran
seq1( + , ((abs β β ) β π)) β β* β§ π β β*)
β (π < sup(ran
seq1( + , ((abs β β ) β π)), β*, < ) β
βπ§ β ran seq1( +
, ((abs β β ) β π))π < π§)) |
185 | 181, 183,
184 | syl2anr 598 |
. . . . . . . 8
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (π < sup(ran seq1( + , ((abs β
β ) β π)),
β*, < ) β βπ§ β ran seq1( + , ((abs β β )
β π))π < π§)) |
186 | 135, 185 | mpbid 231 |
. . . . . . 7
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βπ§ β ran seq1( + , ((abs β β )
β π))π < π§) |
187 | | seqfn 13975 |
. . . . . . . . . 10
β’ (1 β
β€ β seq1( + , ((abs β β ) β π)) Fn
(β€β₯β1)) |
188 | 174, 187 | ax-mp 5 |
. . . . . . . . 9
β’ seq1( + ,
((abs β β ) β π)) Fn
(β€β₯β1) |
189 | 173 | fneq2i 6645 |
. . . . . . . . 9
β’ (seq1( +
, ((abs β β ) β π)) Fn β β seq1( + , ((abs β
β ) β π)) Fn
(β€β₯β1)) |
190 | 188, 189 | mpbir 230 |
. . . . . . . 8
β’ seq1( + ,
((abs β β ) β π)) Fn β |
191 | | breq2 5152 |
. . . . . . . . 9
β’ (π§ = (seq1( + , ((abs β
β ) β π))βπ) β (π < π§ β π < (seq1( + , ((abs β β )
β π))βπ))) |
192 | 191 | rexrn 7086 |
. . . . . . . 8
β’ (seq1( +
, ((abs β β ) β π)) Fn β β (βπ§ β ran seq1( + , ((abs
β β ) β π))π < π§ β βπ β β π < (seq1( + , ((abs β β )
β π))βπ))) |
193 | 190, 192 | ax-mp 5 |
. . . . . . 7
β’
(βπ§ β ran
seq1( + , ((abs β β ) β π))π < π§ β βπ β β π < (seq1( + , ((abs β β )
β π))βπ)) |
194 | 186, 193 | sylib 217 |
. . . . . 6
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βπ β β π < (seq1( + , ((abs β β )
β π))βπ)) |
195 | 60 | ffvelcdmda 7084 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β ( β€ β© (β Γ
β))) |
196 | | 0le0 12310 |
. . . . . . . . . . . . . . . . . 18
β’ 0 β€
0 |
197 | | df-br 5149 |
. . . . . . . . . . . . . . . . . 18
β’ (0 β€ 0
β β¨0, 0β© β β€ ) |
198 | 196, 197 | mpbi 229 |
. . . . . . . . . . . . . . . . 17
β’ β¨0,
0β© β β€ |
199 | | 0re 11213 |
. . . . . . . . . . . . . . . . . 18
β’ 0 β
β |
200 | | opelxpi 5713 |
. . . . . . . . . . . . . . . . . 18
β’ ((0
β β β§ 0 β β) β β¨0, 0β© β (β
Γ β)) |
201 | 199, 199,
200 | mp2an 691 |
. . . . . . . . . . . . . . . . 17
β’ β¨0,
0β© β (β Γ β) |
202 | | elin 3964 |
. . . . . . . . . . . . . . . . 17
β’ (β¨0,
0β© β ( β€ β© (β Γ β)) β (β¨0, 0β©
β β€ β§ β¨0, 0β© β (β Γ
β))) |
203 | 198, 201,
202 | mpbir2an 710 |
. . . . . . . . . . . . . . . 16
β’ β¨0,
0β© β ( β€ β© (β Γ β)) |
204 | | ifcl 4573 |
. . . . . . . . . . . . . . . 16
β’ (((πβπ§) β ( β€ β© (β Γ
β)) β§ β¨0, 0β© β ( β€ β© (β Γ
β))) β if(π§
β (1...π), (πβπ§), β¨0, 0β©) β ( β€ β©
(β Γ β))) |
205 | 195, 203,
204 | sylancl 587 |
. . . . . . . . . . . . . . 15
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) β ( β€ β©
(β Γ β))) |
206 | 205 | fmpttd 7112 |
. . . . . . . . . . . . . 14
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)):ββΆ( β€
β© (β Γ β))) |
207 | | df-ov 7409 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (0(,)0) =
((,)ββ¨0, 0β©) |
208 | | iooid 13349 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (0(,)0) =
β
|
209 | 207, 208 | eqtr3i 2763 |
. . . . . . . . . . . . . . . . . . . . 21
β’
((,)ββ¨0, 0β©) = β
|
210 | 209 | ineq1i 4208 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((,)ββ¨0, 0β©) β© ((,)β(πβπ§))) = (β
β© ((,)β(πβπ§))) |
211 | | 0in 4393 |
. . . . . . . . . . . . . . . . . . . 20
β’ (β
β© ((,)β(πβπ§))) = β
|
212 | 210, 211 | eqtri 2761 |
. . . . . . . . . . . . . . . . . . 19
β’
(((,)ββ¨0, 0β©) β© ((,)β(πβπ§))) = β
|
213 | 212 | olci 865 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π§ β¨ (((,)ββ¨0, 0β©) β©
((,)β(πβπ§))) = β
) |
214 | | ineq1 4205 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(((,)β(πβπ)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
(((,)β(πβπ)) β© ((,)β(πβπ§))) = (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§)))) |
215 | 214 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((,)β(πβπ)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
((((,)β(πβπ)) β© ((,)β(πβπ§))) = β
β (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
)) |
216 | 215 | orbi2d 915 |
. . . . . . . . . . . . . . . . . . 19
β’
(((,)β(πβπ)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
((π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) =
β
))) |
217 | | ineq1 4205 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(((,)ββ¨0, 0β©) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
(((,)ββ¨0, 0β©) β© ((,)β(πβπ§))) = (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§)))) |
218 | 217 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((,)ββ¨0, 0β©) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
((((,)ββ¨0, 0β©) β© ((,)β(πβπ§))) = β
β (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
)) |
219 | 218 | orbi2d 915 |
. . . . . . . . . . . . . . . . . . 19
β’
(((,)ββ¨0, 0β©) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
((π = π§ β¨ (((,)ββ¨0, 0β©) β©
((,)β(πβπ§))) = β
) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) =
β
))) |
220 | 216, 219 | ifboth 4567 |
. . . . . . . . . . . . . . . . . 18
β’ (((π = π§ β¨ (((,)β(πβπ)) β© ((,)β(πβπ§))) = β
) β§ (π = π§ β¨ (((,)ββ¨0, 0β©) β©
((,)β(πβπ§))) = β
)) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
)) |
221 | 112, 213,
220 | sylancl 587 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
)) |
222 | 209 | ineq2i 4209 |
. . . . . . . . . . . . . . . . . . 19
β’ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
β
) |
223 | | in0 4391 |
. . . . . . . . . . . . . . . . . . 19
β’ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
β
) = β
|
224 | 222, 223 | eqtri 2761 |
. . . . . . . . . . . . . . . . . 18
β’ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = β
|
225 | 224 | olci 865 |
. . . . . . . . . . . . . . . . 17
β’ (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = β
) |
226 | | ineq2 4206 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((,)β(πβπ§)) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
(if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0,
0β©)))) |
227 | 226 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . 19
β’
(((,)β(πβπ§)) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
((if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
β (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
228 | 227 | orbi2d 915 |
. . . . . . . . . . . . . . . . . 18
β’
(((,)β(πβπ§)) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
((π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
))) |
229 | | ineq2 4206 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((,)ββ¨0, 0β©) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
(if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0,
0β©)))) |
230 | 229 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . 19
β’
(((,)ββ¨0, 0β©) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
((if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = β
β (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
231 | 230 | orbi2d 915 |
. . . . . . . . . . . . . . . . . 18
β’
(((,)ββ¨0, 0β©) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©)) β
((π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = β
) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
))) |
232 | 228, 231 | ifboth 4567 |
. . . . . . . . . . . . . . . . 17
β’ (((π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)β(πβπ§))) = β
) β§ (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
((,)ββ¨0, 0β©)) = β
)) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
233 | 221, 225,
232 | sylancl 587 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π§ β β)) β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
234 | 233 | ralrimivva 3201 |
. . . . . . . . . . . . . . 15
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ β β βπ§ β β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
235 | | disjeq2 5117 |
. . . . . . . . . . . . . . . . 17
β’
(βπ β
β ((,)β((π§
β β β¦ if(π§
β (1...π), (πβπ§), β¨0, 0β©))βπ)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
(Disj π β
β ((,)β((π§
β β β¦ if(π§
β (1...π), (πβπ§), β¨0, 0β©))βπ)) β Disj π β β if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0,
0β©)))) |
236 | | eleq1w 2817 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π§ = π β (π§ β (1...π) β π β (1...π))) |
237 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π§ = π β (πβπ§) = (πβπ)) |
238 | 236, 237 | ifbieq1d 4552 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π§ = π β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = if(π β (1...π), (πβπ), β¨0, 0β©)) |
239 | | eqid 2733 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) = (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) |
240 | | fvex 6902 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (πβπ) β V |
241 | | opex 5464 |
. . . . . . . . . . . . . . . . . . . . 21
β’ β¨0,
0β© β V |
242 | 240, 241 | ifex 4578 |
. . . . . . . . . . . . . . . . . . . 20
β’ if(π β (1...π), (πβπ), β¨0, 0β©) β
V |
243 | 238, 239,
242 | fvmpt 6996 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β β β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = if(π β (1...π), (πβπ), β¨0, 0β©)) |
244 | 243 | fveq2d 6893 |
. . . . . . . . . . . . . . . . . 18
β’ (π β β β
((,)β((π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ)) = ((,)βif(π β (1...π), (πβπ), β¨0, 0β©))) |
245 | | fvif 6905 |
. . . . . . . . . . . . . . . . . 18
β’
((,)βif(π
β (1...π), (πβπ), β¨0, 0β©)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0,
0β©)) |
246 | 244, 245 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β
((,)β((π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ)) = if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0,
0β©))) |
247 | 235, 246 | mprg 3068 |
. . . . . . . . . . . . . . . 16
β’
(Disj π
β β ((,)β((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ)) β Disj π β β if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0,
0β©))) |
248 | | eleq1w 2817 |
. . . . . . . . . . . . . . . . . 18
β’ (π = π§ β (π β (1...π) β π§ β (1...π))) |
249 | 248, 115 | ifbieq1d 4552 |
. . . . . . . . . . . . . . . . 17
β’ (π = π§ β if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) = if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0,
0β©))) |
250 | 249 | disjor 5128 |
. . . . . . . . . . . . . . . 16
β’
(Disj π
β β if(π β
(1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β
βπ β β
βπ§ β β
(π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
251 | 247, 250 | bitri 275 |
. . . . . . . . . . . . . . 15
β’
(Disj π
β β ((,)β((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ)) β βπ β β βπ§ β β (π = π§ β¨ (if(π β (1...π), ((,)β(πβπ)), ((,)ββ¨0, 0β©)) β©
if(π§ β (1...π), ((,)β(πβπ§)), ((,)ββ¨0, 0β©))) =
β
)) |
252 | 234, 251 | sylibr 233 |
. . . . . . . . . . . . . 14
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β Disj π β β ((,)β((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ))) |
253 | | eqid 2733 |
. . . . . . . . . . . . . 14
β’ seq1( + ,
((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) = seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) |
254 | 206, 252,
253 | uniiccvol 25089 |
. . . . . . . . . . . . 13
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (vol*ββͺ ran ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) = sup(ran seq1( + ,
((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))), β*,
< )) |
255 | 254 | adantr 482 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (vol*ββͺ ran ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) = sup(ran seq1( + ,
((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))), β*,
< )) |
256 | | rexpssxrxp 11256 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (β
Γ β) β (β* Γ
β*) |
257 | 164, 256 | sstri 3991 |
. . . . . . . . . . . . . . . . . . . 20
β’ {π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (β* Γ
β*) |
258 | 257, 65 | sselid 3980 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β (β* Γ
β*)) |
259 | | 0xr 11258 |
. . . . . . . . . . . . . . . . . . . 20
β’ 0 β
β* |
260 | | opelxpi 5713 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((0
β β* β§ 0 β β*) β β¨0,
0β© β (β* Γ
β*)) |
261 | 259, 259,
260 | mp2an 691 |
. . . . . . . . . . . . . . . . . . 19
β’ β¨0,
0β© β (β* Γ
β*) |
262 | | ifcl 4573 |
. . . . . . . . . . . . . . . . . . 19
β’ (((πβπ§) β (β* Γ
β*) β§ β¨0, 0β© β (β* Γ
β*)) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) β
(β* Γ β*)) |
263 | 258, 261,
262 | sylancl 587 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) β
(β* Γ β*)) |
264 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) = (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
265 | | iccf 13422 |
. . . . . . . . . . . . . . . . . . . 20
β’
[,]:(β* Γ β*)βΆπ«
β* |
266 | 265 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β [,]:(β* Γ
β*)βΆπ« β*) |
267 | 266 | feqmptd 6958 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β [,] = (π β (β* Γ
β*) β¦ ([,]βπ))) |
268 | | fveq2 6889 |
. . . . . . . . . . . . . . . . . 18
β’ (π = if(π§ β (1...π), (πβπ§), β¨0, 0β©) β ([,]βπ) = ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
269 | 263, 264,
267, 268 | fmptco 7124 |
. . . . . . . . . . . . . . . . 17
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) = (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©)))) |
270 | 52, 269 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) = (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©)))) |
271 | 270 | rneqd 5936 |
. . . . . . . . . . . . . . 15
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) = ran (π§ β β β¦
([,]βif(π§ β
(1...π), (πβπ§), β¨0, 0β©)))) |
272 | 271 | unieqd 4922 |
. . . . . . . . . . . . . 14
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ ran
([,] β (π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))) = βͺ ran (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©)))) |
273 | | peano2nn 12221 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β β (π + 1) β
β) |
274 | 273, 173 | eleqtrdi 2844 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β β β (π + 1) β
(β€β₯β1)) |
275 | | fzouzsplit 13664 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π + 1) β
(β€β₯β1) β (β€β₯β1) =
((1..^(π + 1)) βͺ
(β€β₯β(π + 1)))) |
276 | 274, 275 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β β β
(β€β₯β1) = ((1..^(π + 1)) βͺ
(β€β₯β(π + 1)))) |
277 | 173, 276 | eqtrid 2785 |
. . . . . . . . . . . . . . . . . 18
β’ (π β β β β =
((1..^(π + 1)) βͺ
(β€β₯β(π + 1)))) |
278 | | nnz 12576 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β β β π β
β€) |
279 | | fzval3 13698 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β β€ β
(1...π) = (1..^(π + 1))) |
280 | 278, 279 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β β β
(1...π) = (1..^(π + 1))) |
281 | 280 | uneq1d 4162 |
. . . . . . . . . . . . . . . . . 18
β’ (π β β β
((1...π) βͺ
(β€β₯β(π + 1))) = ((1..^(π + 1)) βͺ
(β€β₯β(π + 1)))) |
282 | 277, 281 | eqtr4d 2776 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β β =
((1...π) βͺ
(β€β₯β(π + 1)))) |
283 | | fvif 6905 |
. . . . . . . . . . . . . . . . . 18
β’
([,]βif(π§
β (1...π), (πβπ§), β¨0, 0β©)) = if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0,
0β©)) |
284 | 283 | a1i 11 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β
([,]βif(π§ β
(1...π), (πβπ§), β¨0, 0β©)) = if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0,
0β©))) |
285 | 282, 284 | iuneq12d 5025 |
. . . . . . . . . . . . . . . 16
β’ (π β β β βͺ π§ β β ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©)) = βͺ π§ β ((1...π) βͺ (β€β₯β(π + 1)))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0,
0β©))) |
286 | | fvex 6902 |
. . . . . . . . . . . . . . . . 17
β’
([,]βif(π§
β (1...π), (πβπ§), β¨0, 0β©)) β
V |
287 | 286 | dfiun3 5964 |
. . . . . . . . . . . . . . . 16
β’ βͺ π§ β β ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©)) = βͺ ran (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
288 | | iunxun 5097 |
. . . . . . . . . . . . . . . 16
β’ βͺ π§ β ((1...π) βͺ (β€β₯β(π + 1)))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) = (βͺ π§ β (1...π)if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) βͺ
βͺ π§ β (β€β₯β(π + 1))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0,
0β©))) |
289 | 285, 287,
288 | 3eqtr3g 2796 |
. . . . . . . . . . . . . . 15
β’ (π β β β βͺ ran (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©))) = (βͺ π§ β (1...π)if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) βͺ
βͺ π§ β (β€β₯β(π + 1))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0,
0β©)))) |
290 | | iftrue 4534 |
. . . . . . . . . . . . . . . . . 18
β’ (π§ β (1...π) β if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) =
([,]β(πβπ§))) |
291 | 290 | iuneq2i 5018 |
. . . . . . . . . . . . . . . . 17
β’ βͺ π§ β (1...π)if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) = βͺ π§ β (1...π)([,]β(πβπ§)) |
292 | 291 | a1i 11 |
. . . . . . . . . . . . . . . 16
β’ (π β β β βͺ π§ β (1...π)if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) = βͺ π§ β (1...π)([,]β(πβπ§))) |
293 | | uznfz 13581 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π§ β
(β€β₯β(π + 1)) β Β¬ π§ β (1...((π + 1) β 1))) |
294 | 293 | adantl 483 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β β β§ π§ β
(β€β₯β(π + 1))) β Β¬ π§ β (1...((π + 1) β 1))) |
295 | | nncn 12217 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π β β β π β
β) |
296 | | ax-1cn 11165 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ 1 β
β |
297 | | pncan 11463 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β β β§ 1 β
β) β ((π + 1)
β 1) = π) |
298 | 295, 296,
297 | sylancl 587 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β β β ((π + 1) β 1) = π) |
299 | 298 | oveq2d 7422 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β β β
(1...((π + 1) β 1)) =
(1...π)) |
300 | 299 | eleq2d 2820 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β β β (π§ β (1...((π + 1) β 1)) β π§ β (1...π))) |
301 | 300 | notbid 318 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β β β (Β¬
π§ β (1...((π + 1) β 1)) β Β¬
π§ β (1...π))) |
302 | 301 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β β β§ π§ β
(β€β₯β(π + 1))) β (Β¬ π§ β (1...((π + 1) β 1)) β Β¬ π§ β (1...π))) |
303 | 294, 302 | mpbid 231 |
. . . . . . . . . . . . . . . . . 18
β’ ((π β β β§ π§ β
(β€β₯β(π + 1))) β Β¬ π§ β (1...π)) |
304 | 303 | iffalsed 4539 |
. . . . . . . . . . . . . . . . 17
β’ ((π β β β§ π§ β
(β€β₯β(π + 1))) β if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) =
([,]ββ¨0, 0β©)) |
305 | 304 | iuneq2dv 5021 |
. . . . . . . . . . . . . . . 16
β’ (π β β β βͺ π§ β (β€β₯β(π + 1))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) = βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©)) |
306 | 292, 305 | uneq12d 4164 |
. . . . . . . . . . . . . . 15
β’ (π β β β (βͺ π§ β (1...π)if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©)) βͺ
βͺ π§ β (β€β₯β(π + 1))if(π§ β (1...π), ([,]β(πβπ§)), ([,]ββ¨0, 0β©))) =
(βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0,
0β©))) |
307 | 289, 306 | eqtrd 2773 |
. . . . . . . . . . . . . 14
β’ (π β β β βͺ ran (π§ β β β¦ ([,]βif(π§ β (1...π), (πβπ§), β¨0, 0β©))) = (βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0,
0β©))) |
308 | 272, 307 | sylan9eq 2793 |
. . . . . . . . . . . . 13
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β βͺ ran ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) = (βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0,
0β©))) |
309 | 308 | fveq2d 6893 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (vol*ββͺ ran ([,] β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) =
(vol*β(βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0,
0β©)))) |
310 | | xrltso 13117 |
. . . . . . . . . . . . . . 15
β’ < Or
β* |
311 | 310 | a1i 11 |
. . . . . . . . . . . . . 14
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β < Or
β*) |
312 | | elnnuz 12863 |
. . . . . . . . . . . . . . . . . 18
β’ (π β β β π β
(β€β₯β1)) |
313 | 312 | biimpi 215 |
. . . . . . . . . . . . . . . . 17
β’ (π β β β π β
(β€β₯β1)) |
314 | 313 | adantl 483 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β π β
(β€β₯β1)) |
315 | | elfznn 13527 |
. . . . . . . . . . . . . . . . . 18
β’ (π’ β (1...π) β π’ β β) |
316 | 172 | ffvelcdmda 7084 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π’ β β) β (((abs β
β ) β π)βπ’) β β) |
317 | 315, 316 | sylan2 594 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π’ β (1...π)) β (((abs β β ) β
π)βπ’) β β) |
318 | 317 | adantlr 714 |
. . . . . . . . . . . . . . . 16
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π’ β (1...π)) β (((abs β β ) β
π)βπ’) β β) |
319 | | readdcl 11190 |
. . . . . . . . . . . . . . . . 17
β’ ((π’ β β β§ π£ β β) β (π’ + π£) β β) |
320 | 319 | adantl 483 |
. . . . . . . . . . . . . . . 16
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ (π’ β β β§ π£ β β)) β (π’ + π£) β β) |
321 | 314, 318,
320 | seqcl 13985 |
. . . . . . . . . . . . . . 15
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) β β) |
322 | 321 | rexrd 11261 |
. . . . . . . . . . . . . 14
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) β
β*) |
323 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β (1...π) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) = (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
324 | | iftrue 4534 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π β (1...π) β if(π β (1...π), (πβπ), β¨0, 0β©) = (πβπ)) |
325 | 238, 324 | sylan9eqr 2795 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π β (1...π) β§ π§ = π) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = (πβπ)) |
326 | | elfznn 13527 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β (1...π) β π β β) |
327 | 240 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π β (1...π) β (πβπ) β V) |
328 | 323, 325,
326, 327 | fvmptd 7003 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π β (1...π) β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = (πβπ)) |
329 | 328 | adantl 483 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = (πβπ)) |
330 | 329 | fveq2d 6893 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ)) = ((abs β β
)β(πβπ))) |
331 | | fvex 6902 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (πβπ§) β V |
332 | 331, 241 | ifex 4578 |
. . . . . . . . . . . . . . . . . . . . 21
β’ if(π§ β (1...π), (πβπ§), β¨0, 0β©) β
V |
333 | 332, 239 | fnmpti 6691 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) Fn
β |
334 | | fvco2 6986 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) Fn β β§ π β β) β (((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
335 | 333, 326,
334 | sylancr 588 |
. . . . . . . . . . . . . . . . . . 19
β’ (π β (1...π) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
336 | 335 | adantl 483 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
337 | | ffn 6715 |
. . . . . . . . . . . . . . . . . . 19
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β π Fn β) |
338 | | fvco2 6986 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π Fn β β§ π β β) β (((abs
β β ) β π)βπ) = ((abs β β )β(πβπ))) |
339 | 337, 326,
338 | syl2an 597 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β (((abs β β ) β
π)βπ) = ((abs β β )β(πβπ))) |
340 | 330, 336,
339 | 3eqtr4d 2783 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = (((abs β β )
β π)βπ)) |
341 | 340 | adantlr 714 |
. . . . . . . . . . . . . . . 16
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β (1...π)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = (((abs β β )
β π)βπ)) |
342 | 314, 341 | seqfveq 13989 |
. . . . . . . . . . . . . . 15
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ) = (seq1( + , ((abs β
β ) β π))βπ)) |
343 | 174 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β 1 β β€) |
344 | 168, 65 | sselid 3980 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β (β Γ
β)) |
345 | | 0cn 11203 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ 0 β
β |
346 | | opelxpi 5713 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((0
β β β§ 0 β β) β β¨0, 0β© β (β
Γ β)) |
347 | 345, 345,
346 | mp2an 691 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ β¨0,
0β© β (β Γ β) |
348 | | ifcl 4573 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((πβπ§) β (β Γ β) β§
β¨0, 0β© β (β Γ β)) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) β (β Γ
β)) |
349 | 344, 347,
348 | sylancl 587 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) β (β Γ
β)) |
350 | 349 | fmpttd 7112 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)):ββΆ(β
Γ β)) |
351 | | fco 6739 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((abs
β β ):(β Γ β)βΆβ β§ (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)):ββΆ(β
Γ β)) β ((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0,
0β©))):ββΆβ) |
352 | 139, 350,
351 | sylancr 588 |
. . . . . . . . . . . . . . . . . . 19
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0,
0β©))):ββΆβ) |
353 | 352 | ffvelcdmda 7084 |
. . . . . . . . . . . . . . . . . 18
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (((abs β
β ) β (π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©)))βπ) β
β) |
354 | 173, 343,
353 | serfre 13994 |
. . . . . . . . . . . . . . . . 17
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0,
0β©)))):ββΆβ) |
355 | 354 | ffnd 6716 |
. . . . . . . . . . . . . . . 16
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©)))) Fn
β) |
356 | | fnfvelrn 7080 |
. . . . . . . . . . . . . . . 16
β’ ((seq1( +
, ((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) Fn β β§
π β β) β
(seq1( + , ((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ) β ran seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))) |
357 | 355, 356 | sylan 581 |
. . . . . . . . . . . . . . 15
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ) β ran seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))) |
358 | 342, 357 | eqeltrrd 2835 |
. . . . . . . . . . . . . 14
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) β ran seq1( + , ((abs β β
) β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))) |
359 | 354 | frnd 6723 |
. . . . . . . . . . . . . . . . 17
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β ran seq1( + , ((abs β β
) β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©)))) β
β) |
360 | 359 | adantr 482 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β ran seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) β
β) |
361 | 360 | sselda 3982 |
. . . . . . . . . . . . . . 15
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β ran seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))) β π β
β) |
362 | 321 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β ran seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))) β (seq1( + ,
((abs β β ) β π))βπ) β β) |
363 | | readdcl 11190 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β β§ π’ β β) β (π + π’) β β) |
364 | 363 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β§ (π β β β§ π’ β β)) β (π + π’) β β) |
365 | | recn 11197 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π β β β π β
β) |
366 | | recn 11197 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π’ β β β π’ β
β) |
367 | | recn 11197 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (π£ β β β π£ β
β) |
368 | | addass 11194 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β β β§ π’ β β β§ π£ β β) β ((π + π’) + π£) = (π + (π’ + π£))) |
369 | 365, 366,
367, 368 | syl3an 1161 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β β§ π’ β β β§ π£ β β) β ((π + π’) + π£) = (π + (π’ + π£))) |
370 | 369 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β§ (π β β β§ π’ β β β§ π£ β β)) β ((π + π’) + π£) = (π + (π’ + π£))) |
371 | | nnltp1le 12615 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β β β§ π‘ β β) β (π < π‘ β (π + 1) β€ π‘)) |
372 | 371 | biimpa 478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π β β β§ π‘ β β) β§ π < π‘) β (π + 1) β€ π‘) |
373 | 273 | nnzd 12582 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β β β (π + 1) β
β€) |
374 | | nnz 12576 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π‘ β β β π‘ β
β€) |
375 | | eluz 12833 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (((π + 1) β β€ β§ π‘ β β€) β (π‘ β
(β€β₯β(π + 1)) β (π + 1) β€ π‘)) |
376 | 373, 374,
375 | syl2an 597 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π β β β§ π‘ β β) β (π‘ β
(β€β₯β(π + 1)) β (π + 1) β€ π‘)) |
377 | 376 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π β β β§ π‘ β β) β§ π < π‘) β (π‘ β (β€β₯β(π + 1)) β (π + 1) β€ π‘)) |
378 | 372, 377 | mpbird 257 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β β β§ π‘ β β) β§ π < π‘) β π‘ β (β€β₯β(π + 1))) |
379 | 378 | adantlll 717 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β π‘ β (β€β₯β(π + 1))) |
380 | 313 | ad3antlr 730 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β π β
(β€β₯β1)) |
381 | | simplll 774 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
382 | | elfznn 13527 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β (1...π‘) β π β β) |
383 | 381, 382,
353 | syl2an 597 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β§ π β (1...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) β
β) |
384 | 364, 370,
379, 380, 383 | seqsplit 13998 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) = ((seq1( + , ((abs β
β ) β (π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ) + (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘))) |
385 | 342 | ad2antrr 725 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ) = (seq1( + , ((abs β
β ) β π))βπ)) |
386 | | elfzelz 13498 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (π β ((π + 1)...π‘) β π β β€) |
387 | 386 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β π β β€) |
388 | | 0red 11214 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β 0 β β) |
389 | 273 | nnred 12224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ (π β β β (π + 1) β
β) |
390 | 389 | ad3antrrr 729 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (π + 1) β β) |
391 | 386 | zred 12663 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ (π β ((π + 1)...π‘) β π β β) |
392 | 391 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β π β β) |
393 | 273 | nngt0d 12258 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ (π β β β 0 <
(π + 1)) |
394 | 393 | ad3antrrr 729 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β 0 < (π + 1)) |
395 | | elfzle1 13501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ (π β ((π + 1)...π‘) β (π + 1) β€ π) |
396 | 395 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (π + 1) β€ π) |
397 | 388, 390,
392, 394, 396 | ltletrd 11371 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β 0 < π) |
398 | | elnnz 12565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (π β β β (π β β€ β§ 0 <
π)) |
399 | 387, 397,
398 | sylanbrc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β π β β) |
400 | 333, 399,
334 | sylancr 588 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
401 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((π β β β§ π β ((π + 1)...π‘)) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) = (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
402 | | nnre 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
β’ (π β β β π β
β) |
403 | 402 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ ((π β β β§ π β ((π + 1)...π‘)) β π β β) |
404 | 389 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ ((π β β β§ π β ((π + 1)...π‘)) β (π + 1) β β) |
405 | 391 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ ((π β β β§ π β ((π + 1)...π‘)) β π β β) |
406 | 402 | ltp1d 12141 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
β’ (π β β β π < (π + 1)) |
407 | 406 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ ((π β β β§ π β ((π + 1)...π‘)) β π < (π + 1)) |
408 | 395 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ ((π β β β§ π β ((π + 1)...π‘)) β (π + 1) β€ π) |
409 | 403, 404,
405, 407, 408 | ltletrd 11371 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
β’ ((π β β β§ π β ((π + 1)...π‘)) β π < π) |
410 | 409 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
β’ (((π β β β§ π β ((π + 1)...π‘)) β§ π§ = π) β π < π) |
411 | 403, 405 | ltnled 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
β’ ((π β β β§ π β ((π + 1)...π‘)) β (π < π β Β¬ π β€ π)) |
412 | | breq1 5151 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
β’ (π = π§ β (π β€ π β π§ β€ π)) |
413 | 412 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
β’ (π§ = π β (π β€ π β π§ β€ π)) |
414 | 413 | notbid 318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
β’ (π§ = π β (Β¬ π β€ π β Β¬ π§ β€ π)) |
415 | 411, 414 | sylan9bb 511 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
β’ (((π β β β§ π β ((π + 1)...π‘)) β§ π§ = π) β (π < π β Β¬ π§ β€ π)) |
416 | 410, 415 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
β’ (((π β β β§ π β ((π + 1)...π‘)) β§ π§ = π) β Β¬ π§ β€ π) |
417 | | elfzle2 13502 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
β’ (π§ β (1...π) β π§ β€ π) |
418 | 416, 417 | nsyl 140 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ (((π β β β§ π β ((π + 1)...π‘)) β§ π§ = π) β Β¬ π§ β (1...π)) |
419 | 418 | iffalsed 4539 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (((π β β β§ π β ((π + 1)...π‘)) β§ π§ = π) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = β¨0,
0β©) |
420 | 386 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ ((π β β β§ π β ((π + 1)...π‘)) β π β β€) |
421 | | 0red 11214 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
β’ ((π β β β§ π β ((π + 1)...π‘)) β 0 β β) |
422 | 393 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
β’ ((π β β β§ π β ((π + 1)...π‘)) β 0 < (π + 1)) |
423 | 421, 404,
405, 422, 408 | ltletrd 11371 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
β’ ((π β β β§ π β ((π + 1)...π‘)) β 0 < π) |
424 | 420, 423,
398 | sylanbrc 584 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((π β β β§ π β ((π + 1)...π‘)) β π β β) |
425 | 241 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((π β β β§ π β ((π + 1)...π‘)) β β¨0, 0β© β
V) |
426 | 401, 419,
424, 425 | fvmptd 7003 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((π β β β§ π β ((π + 1)...π‘)) β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = β¨0,
0β©) |
427 | 426 | ad4ant14 751 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = β¨0,
0β©) |
428 | 427 | fveq2d 6893 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ)) = ((abs β β
)ββ¨0, 0β©)) |
429 | 400, 428 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)ββ¨0, 0β©)) |
430 | | fvco3 6988 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((
β :(β Γ β)βΆβ β§ β¨0, 0β© β
(β Γ β)) β ((abs β β )ββ¨0,
0β©) = (absβ( β ββ¨0, 0β©))) |
431 | 137, 347,
430 | mp2an 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((abs
β β )ββ¨0, 0β©) = (absβ( β
ββ¨0, 0β©)) |
432 | | df-ov 7409 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (0
β 0) = ( β ββ¨0, 0β©) |
433 | | 0m0e0 12329 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (0
β 0) = 0 |
434 | 432, 433 | eqtr3i 2763 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ( β
ββ¨0, 0β©) = 0 |
435 | 434 | fveq2i 6892 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’
(absβ( β ββ¨0, 0β©)) =
(absβ0) |
436 | | abs0 15229 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’
(absβ0) = 0 |
437 | 435, 436 | eqtri 2761 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’
(absβ( β ββ¨0, 0β©)) = 0 |
438 | 431, 437 | eqtri 2761 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ ((abs
β β )ββ¨0, 0β©) = 0 |
439 | 429, 438 | eqtrdi 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = 0) |
440 | | elfzuz 13494 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β ((π + 1)...π‘) β π β (β€β₯β(π + 1))) |
441 | | c0ex 11205 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ 0 β
V |
442 | 441 | fvconst2 7202 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ (π β
(β€β₯β(π + 1)) β
(((β€β₯β(π + 1)) Γ {0})βπ) = 0) |
443 | 440, 442 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π β ((π + 1)...π‘) β
(((β€β₯β(π + 1)) Γ {0})βπ) = 0) |
444 | 443 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β
(((β€β₯β(π + 1)) Γ {0})βπ) = 0) |
445 | 439, 444 | eqtr4d 2776 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((((π β β β§ π‘ β β) β§ π < π‘) β§ π β ((π + 1)...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) =
(((β€β₯β(π + 1)) Γ {0})βπ)) |
446 | 378, 445 | seqfveq 13989 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (((π β β β§ π‘ β β) β§ π < π‘) β (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) = (seq(π + 1)( + ,
((β€β₯β(π + 1)) Γ {0}))βπ‘)) |
447 | | eqid 2733 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
(β€β₯β(π + 1)) = (β€β₯β(π + 1)) |
448 | 447 | ser0 14017 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π‘ β
(β€β₯β(π + 1)) β (seq(π + 1)( + ,
((β€β₯β(π + 1)) Γ {0}))βπ‘) = 0) |
449 | 378, 448 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (((π β β β§ π‘ β β) β§ π < π‘) β (seq(π + 1)( + ,
((β€β₯β(π + 1)) Γ {0}))βπ‘) = 0) |
450 | 446, 449 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π β β β§ π‘ β β) β§ π < π‘) β (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) = 0) |
451 | 450 | adantlll 717 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) = 0) |
452 | 385, 451 | oveq12d 7424 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β ((seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ) + (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘)) = ((seq1( + , ((abs β
β ) β π))βπ) + 0)) |
453 | 172 | ffvelcdmda 7084 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (((abs β
β ) β π)βπ) β β) |
454 | 326, 453 | sylan2 594 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β (1...π)) β (((abs β β ) β
π)βπ) β β) |
455 | 454 | adantlr 714 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β (1...π)) β (((abs β β ) β
π)βπ) β β) |
456 | | readdcl 11190 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ ((π β β β§ π£ β β) β (π + π£) β β) |
457 | 456 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ (π β β β§ π£ β β)) β (π + π£) β β) |
458 | 314, 455,
457 | seqcl 13985 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) β β) |
459 | 458 | ad2antrr 725 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β π))βπ) β
β) |
460 | 459 | recnd 11239 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β π))βπ) β
β) |
461 | 460 | addridd 11411 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β ((seq1( + , ((abs β β )
β π))βπ) + 0) = (seq1( + , ((abs
β β ) β π))βπ)) |
462 | 452, 461 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β ((seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ) + (seq(π + 1)( + , ((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘)) = (seq1( + , ((abs β
β ) β π))βπ)) |
463 | 384, 462 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) = (seq1( + , ((abs β
β ) β π))βπ)) |
464 | 453 | ad5ant15 758 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β§ π β β) β (((abs β
β ) β π)βπ) β β) |
465 | 326, 464 | sylan2 594 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β§ π β (1...π)) β (((abs β β ) β
π)βπ) β β) |
466 | 380, 465,
364 | seqcl 13985 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β π))βπ) β
β) |
467 | 466 | leidd 11777 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β π))βπ) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
468 | 463, 467 | eqbrtrd 5170 |
. . . . . . . . . . . . . . . . . . 19
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π < π‘) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
469 | | elnnuz 12863 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π‘ β β β π‘ β
(β€β₯β1)) |
470 | 469 | biimpi 215 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (π‘ β β β π‘ β
(β€β₯β1)) |
471 | 470 | ad2antlr 726 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β π‘ β
(β€β₯β1)) |
472 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)) = (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))) |
473 | | simpr 486 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β π§ = π) |
474 | | elfzle1 13501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (π β (1...π‘) β 1 β€ π) |
475 | 474 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β 1 β€ π) |
476 | 382 | nnred 12224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (π β (1...π‘) β π β β) |
477 | 476 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β β) |
478 | | nnre 12216 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (π‘ β β β π‘ β
β) |
479 | 478 | ad3antlr 730 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π‘ β β) |
480 | 402 | ad3antrrr 729 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β β) |
481 | | elfzle2 13502 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ (π β (1...π‘) β π β€ π‘) |
482 | 481 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β€ π‘) |
483 | | simplr 768 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π‘ β€ π) |
484 | 477, 479,
480, 482, 483 | letrd 11368 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β€ π) |
485 | | elfzelz 13498 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (π β (1...π‘) β π β β€) |
486 | 278 | ad2antrr 725 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ (((π β β β§ π‘ β β) β§ π‘ β€ π) β π β β€) |
487 | | elfz 13487 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
β’ ((π β β€ β§ 1 β
β€ β§ π β
β€) β (π β
(1...π) β (1 β€
π β§ π β€ π))) |
488 | 174, 487 | mp3an2 1450 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
β’ ((π β β€ β§ π β β€) β (π β (1...π) β (1 β€ π β§ π β€ π))) |
489 | 485, 486,
488 | syl2anr 598 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (π β (1...π) β (1 β€ π β§ π β€ π))) |
490 | 475, 484,
489 | mpbir2and 712 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
β’ ((((π β β β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β (1...π)) |
491 | 490 | ad5ant2345 1371 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β (1...π)) |
492 | 491 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β π β (1...π)) |
493 | 473, 492 | eqeltrd 2834 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β π§ β (1...π)) |
494 | | iftrue 4534 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π§ β (1...π) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = (πβπ§)) |
495 | 493, 494 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = (πβπ§)) |
496 | 237 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β (πβπ§) = (πβπ)) |
497 | 495, 496 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’
((((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β§ π§ = π) β if(π§ β (1...π), (πβπ§), β¨0, 0β©) = (πβπ)) |
498 | 382 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β π β β) |
499 | 240 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (πβπ) β V) |
500 | 472, 497,
498, 499 | fvmptd 7003 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β ((π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))βπ) = (πβπ)) |
501 | 500 | fveq2d 6893 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ)) = ((abs β β
)β(πβπ))) |
502 | 333, 382,
334 | sylancr 588 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β (1...π‘) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
503 | 502 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = ((abs β β
)β((π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))βπ))) |
504 | | simplll 774 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) |
505 | | fvco3 6988 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (((abs β
β ) β π)βπ) = ((abs β β )β(πβπ))) |
506 | 504, 382,
505 | syl2an 597 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (((abs β β ) β
π)βπ) = ((abs β β )β(πβπ))) |
507 | 501, 503,
506 | 3eqtr4d 2783 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π‘)) β (((abs β β ) β
(π§ β β β¦
if(π§ β (1...π), (πβπ§), β¨0, 0β©)))βπ) = (((abs β β )
β π)βπ)) |
508 | 471, 507 | seqfveq 13989 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) = (seq1( + , ((abs β
β ) β π))βπ‘)) |
509 | | eluz 12833 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π‘ β β€ β§ π β β€) β (π β
(β€β₯βπ‘) β π‘ β€ π)) |
510 | 374, 278,
509 | syl2anr 598 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π β β β§ π‘ β β) β (π β
(β€β₯βπ‘) β π‘ β€ π)) |
511 | 510 | biimpar 479 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β β β§ π‘ β β) β§ π‘ β€ π) β π β (β€β₯βπ‘)) |
512 | 511 | adantlll 717 |
. . . . . . . . . . . . . . . . . . . . 21
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β π β (β€β₯βπ‘)) |
513 | 504, 326,
453 | syl2an 597 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β (1...π)) β (((abs β β ) β
π)βπ) β β) |
514 | | elfzelz 13498 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ (π β ((π‘ + 1)...π) β π β β€) |
515 | 514 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β π β β€) |
516 | | 0red 11214 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β 0 β β) |
517 | | peano2nn 12221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
β’ (π‘ β β β (π‘ + 1) β
β) |
518 | 517 | nnred 12224 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π‘ β β β (π‘ + 1) β
β) |
519 | 518 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β (π‘ + 1) β β) |
520 | 514 | zred 12663 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β ((π‘ + 1)...π) β π β β) |
521 | 520 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β π β β) |
522 | 517 | nngt0d 12258 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π‘ β β β 0 <
(π‘ + 1)) |
523 | 522 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β 0 < (π‘ + 1)) |
524 | | elfzle1 13501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
β’ (π β ((π‘ + 1)...π) β (π‘ + 1) β€ π) |
525 | 524 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β (π‘ + 1) β€ π) |
526 | 516, 519,
521, 523, 525 | ltletrd 11371 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β 0 < π) |
527 | 515, 526,
398 | sylanbrc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π‘ β β β§ π β ((π‘ + 1)...π)) β π β β) |
528 | 527 | adantlr 714 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π‘ β β β§ π‘ β€ π) β§ π β ((π‘ + 1)...π)) β π β β) |
529 | 528 | adantlll 717 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β ((π‘ + 1)...π)) β π β β) |
530 | 170 | ffvelcdmda 7084 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (πβπ) β (β Γ
β)) |
531 | | ffvelcdm 7081 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((
β :(β Γ β)βΆβ β§ (πβπ) β (β Γ β)) β (
β β(πβπ)) β β) |
532 | 137, 530,
531 | sylancr 588 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β ( β
β(πβπ)) β
β) |
533 | 532 | absge0d 15388 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β 0 β€ (absβ(
β β(πβπ)))) |
534 | | fvco3 6988 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
β’ ((
β :(β Γ β)βΆβ β§ (πβπ) β (β Γ β)) β
((abs β β )β(πβπ)) = (absβ( β β(πβπ)))) |
535 | 137, 530,
534 | sylancr 588 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β ((abs β β
)β(πβπ)) = (absβ( β
β(πβπ)))) |
536 | 505, 535 | eqtrd 2773 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (((abs β
β ) β π)βπ) = (absβ( β β(πβπ)))) |
537 | 533, 536 | breqtrrd 5176 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β 0 β€ (((abs β
β ) β π)βπ)) |
538 | 537 | ad5ant15 758 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β β) β 0 β€ (((abs β
β ) β π)βπ)) |
539 | 529, 538 | syldan 592 |
. . . . . . . . . . . . . . . . . . . . 21
β’
(((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β§ π β ((π‘ + 1)...π)) β 0 β€ (((abs β β )
β π)βπ)) |
540 | 471, 512,
513, 539 | sermono 13997 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β (seq1( + , ((abs β β )
β π))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
541 | 508, 540 | eqbrtrd 5170 |
. . . . . . . . . . . . . . . . . . 19
β’ ((((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β§ π‘ β€ π) β (seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
542 | 402 | ad2antlr 726 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β π β β) |
543 | 478 | adantl 483 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β π‘ β β) |
544 | 468, 541,
542, 543 | ltlecasei 11319 |
. . . . . . . . . . . . . . . . . 18
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π‘ β β) β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
545 | 544 | ralrimiva 3147 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β βπ‘ β β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ)) |
546 | | breq1 5151 |
. . . . . . . . . . . . . . . . . . . 20
β’ (π = (seq1( + , ((abs β
β ) β (π§ β
β β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))βπ‘) β (π β€ (seq1( + , ((abs β β )
β π))βπ) β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ))) |
547 | 546 | ralrn 7087 |
. . . . . . . . . . . . . . . . . . 19
β’ (seq1( +
, ((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))) Fn β β
(βπ β ran seq1(
+ , ((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))π β€ (seq1( + , ((abs β β )
β π))βπ) β βπ‘ β β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ))) |
548 | 355, 547 | syl 17 |
. . . . . . . . . . . . . . . . . 18
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (βπ β ran seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))π β€ (seq1( + , ((abs β β )
β π))βπ) β βπ‘ β β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ))) |
549 | 548 | adantr 482 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (βπ β ran seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))π β€ (seq1( + , ((abs β β )
β π))βπ) β βπ‘ β β (seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))βπ‘) β€ (seq1( + , ((abs β
β ) β π))βπ))) |
550 | 545, 549 | mpbird 257 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β βπ β ran seq1( + , ((abs
β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©))))π β€ (seq1( + , ((abs β β )
β π))βπ)) |
551 | 550 | r19.21bi 3249 |
. . . . . . . . . . . . . . 15
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β ran seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))) β π β€ (seq1( + , ((abs β
β ) β π))βπ)) |
552 | 361, 362,
551 | lensymd 11362 |
. . . . . . . . . . . . . 14
β’ (((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β§ π β ran seq1( + , ((abs β β )
β (π§ β β
β¦ if(π§ β
(1...π), (πβπ§), β¨0, 0β©))))) β Β¬ (seq1(
+ , ((abs β β ) β π))βπ) < π) |
553 | 311, 322,
358, 552 | supmax 9459 |
. . . . . . . . . . . . 13
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β sup(ran seq1( + ,
((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))), β*,
< ) = (seq1( + , ((abs β β ) β π))βπ)) |
554 | 52, 553 | sylan 581 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β sup(ran seq1( + ,
((abs β β ) β (π§ β β β¦ if(π§ β (1...π), (πβπ§), β¨0, 0β©)))), β*,
< ) = (seq1( + , ((abs β β ) β π))βπ)) |
555 | 255, 309,
554 | 3eqtr3rd 2782 |
. . . . . . . . . . 11
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) = (vol*β(βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0,
0β©)))) |
556 | | elfznn 13527 |
. . . . . . . . . . . . . . . 16
β’ (π§ β (1...π) β π§ β β) |
557 | 164, 65 | sselid 3980 |
. . . . . . . . . . . . . . . . 17
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β (πβπ§) β (β Γ
β)) |
558 | | 1st2nd2 8011 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((πβπ§) β (β Γ β) β
(πβπ§) = β¨(1st β(πβπ§)), (2nd β(πβπ§))β©) |
559 | 558 | fveq2d 6893 |
. . . . . . . . . . . . . . . . . . 19
β’ ((πβπ§) β (β Γ β) β
([,]β(πβπ§)) =
([,]ββ¨(1st β(πβπ§)), (2nd β(πβπ§))β©)) |
560 | | df-ov 7409 |
. . . . . . . . . . . . . . . . . . 19
β’
((1st β(πβπ§))[,](2nd β(πβπ§))) = ([,]ββ¨(1st
β(πβπ§)), (2nd
β(πβπ§))β©) |
561 | 559, 560 | eqtr4di 2791 |
. . . . . . . . . . . . . . . . . 18
β’ ((πβπ§) β (β Γ β) β
([,]β(πβπ§)) = ((1st
β(πβπ§))[,](2nd
β(πβπ§)))) |
562 | | xp1st 8004 |
. . . . . . . . . . . . . . . . . . 19
β’ ((πβπ§) β (β Γ β) β
(1st β(πβπ§)) β β) |
563 | | xp2nd 8005 |
. . . . . . . . . . . . . . . . . . 19
β’ ((πβπ§) β (β Γ β) β
(2nd β(πβπ§)) β β) |
564 | | iccssre 13403 |
. . . . . . . . . . . . . . . . . . 19
β’
(((1st β(πβπ§)) β β β§ (2nd
β(πβπ§)) β β) β
((1st β(πβπ§))[,](2nd β(πβπ§))) β β) |
565 | 562, 563,
564 | syl2anc 585 |
. . . . . . . . . . . . . . . . . 18
β’ ((πβπ§) β (β Γ β) β
((1st β(πβπ§))[,](2nd β(πβπ§))) β β) |
566 | 561, 565 | eqsstrd 4020 |
. . . . . . . . . . . . . . . . 17
β’ ((πβπ§) β (β Γ β) β
([,]β(πβπ§)) β
β) |
567 | 557, 566 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β ([,]β(πβπ§)) β β) |
568 | 52, 556, 567 | syl2an 597 |
. . . . . . . . . . . . . . 15
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β (1...π)) β ([,]β(πβπ§)) β β) |
569 | 568 | ralrimiva 3147 |
. . . . . . . . . . . . . 14
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ§ β (1...π)([,]β(πβπ§)) β β) |
570 | | iunss 5048 |
. . . . . . . . . . . . . 14
β’ (βͺ π§ β (1...π)([,]β(πβπ§)) β β β βπ§ β (1...π)([,]β(πβπ§)) β β) |
571 | 569, 570 | sylibr 233 |
. . . . . . . . . . . . 13
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ
π§ β (1...π)([,]β(πβπ§)) β β) |
572 | 571 | adantr 482 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β βͺ π§ β (1...π)([,]β(πβπ§)) β β) |
573 | | uzid 12834 |
. . . . . . . . . . . . . . . 16
β’ ((π + 1) β β€ β
(π + 1) β
(β€β₯β(π + 1))) |
574 | | ne0i 4334 |
. . . . . . . . . . . . . . . 16
β’ ((π + 1) β
(β€β₯β(π + 1)) β
(β€β₯β(π + 1)) β β
) |
575 | | iunconst 5006 |
. . . . . . . . . . . . . . . 16
β’
((β€β₯β(π + 1)) β β
β βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) = ([,]ββ¨0, 0β©)) |
576 | 373, 573,
574, 575 | 4syl 19 |
. . . . . . . . . . . . . . 15
β’ (π β β β βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) = ([,]ββ¨0, 0β©)) |
577 | | iccid 13366 |
. . . . . . . . . . . . . . . . 17
β’ (0 β
β* β (0[,]0) = {0}) |
578 | 259, 577 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
β’ (0[,]0) =
{0} |
579 | | df-ov 7409 |
. . . . . . . . . . . . . . . 16
β’ (0[,]0) =
([,]ββ¨0, 0β©) |
580 | 578, 579 | eqtr3i 2763 |
. . . . . . . . . . . . . . 15
β’ {0} =
([,]ββ¨0, 0β©) |
581 | 576, 580 | eqtr4di 2791 |
. . . . . . . . . . . . . 14
β’ (π β β β βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) = {0}) |
582 | | snssi 4811 |
. . . . . . . . . . . . . . 15
β’ (0 β
β β {0} β β) |
583 | 199, 582 | ax-mp 5 |
. . . . . . . . . . . . . 14
β’ {0}
β β |
584 | 581, 583 | eqsstrdi 4036 |
. . . . . . . . . . . . 13
β’ (π β β β βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) β β) |
585 | 584 | adantl 483 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) β β) |
586 | 581 | fveq2d 6893 |
. . . . . . . . . . . . . 14
β’ (π β β β
(vol*ββͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©)) = (vol*β{0})) |
587 | 586 | adantl 483 |
. . . . . . . . . . . . 13
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (vol*ββͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©)) = (vol*β{0})) |
588 | | ovolsn 25004 |
. . . . . . . . . . . . . 14
β’ (0 β
β β (vol*β{0}) = 0) |
589 | 199, 588 | ax-mp 5 |
. . . . . . . . . . . . 13
β’
(vol*β{0}) = 0 |
590 | 587, 589 | eqtrdi 2789 |
. . . . . . . . . . . 12
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (vol*ββͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©)) = 0) |
591 | | ovolunnul 25009 |
. . . . . . . . . . . 12
β’
((βͺ π§ β (1...π)([,]β(πβπ§)) β β β§ βͺ π§ β (β€β₯β(π + 1))([,]ββ¨0,
0β©) β β β§ (vol*ββͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0, 0β©)) = 0)
β (vol*β(βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0, 0β©))) =
(vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
592 | 572, 585,
590, 591 | syl3anc 1372 |
. . . . . . . . . . 11
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (vol*β(βͺ π§ β (1...π)([,]β(πβπ§)) βͺ βͺ
π§ β
(β€β₯β(π + 1))([,]ββ¨0, 0β©))) =
(vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
593 | 555, 592 | eqtrd 2773 |
. . . . . . . . . 10
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (seq1( + , ((abs
β β ) β π))βπ) = (vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
594 | 593 | breq2d 5160 |
. . . . . . . . 9
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (π < (seq1( + , ((abs β β )
β π))βπ) β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) |
595 | 594 | biimpd 228 |
. . . . . . . 8
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π β β) β (π < (seq1( + , ((abs β β )
β π))βπ) β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) |
596 | 595 | reximdva 3169 |
. . . . . . 7
β’ (π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β (βπ β β π < (seq1( + , ((abs β β )
β π))βπ) β βπ β β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) |
597 | 596 | adantl 483 |
. . . . . 6
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β (βπ β β π < (seq1( + , ((abs β β )
β π))βπ) β βπ β β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) |
598 | 194, 597 | mpd 15 |
. . . . 5
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βπ β β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
599 | | fzfi 13934 |
. . . . . . . . . 10
β’
(1...π) β
Fin |
600 | | icccld 24275 |
. . . . . . . . . . . . . . 15
β’
(((1st β(πβπ§)) β β β§ (2nd
β(πβπ§)) β β) β
((1st β(πβπ§))[,](2nd β(πβπ§))) β (Clsdβ(topGenβran
(,)))) |
601 | 562, 563,
600 | syl2anc 585 |
. . . . . . . . . . . . . 14
β’ ((πβπ§) β (β Γ β) β
((1st β(πβπ§))[,](2nd β(πβπ§))) β (Clsdβ(topGenβran
(,)))) |
602 | 561, 601 | eqeltrd 2834 |
. . . . . . . . . . . . 13
β’ ((πβπ§) β (β Γ β) β
([,]β(πβπ§)) β
(Clsdβ(topGenβran (,)))) |
603 | 557, 602 | syl 17 |
. . . . . . . . . . . 12
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β ([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
604 | 556, 603 | sylan2 594 |
. . . . . . . . . . 11
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β (1...π)) β ([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
605 | 604 | ralrimiva 3147 |
. . . . . . . . . 10
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
606 | | uniretop 24271 |
. . . . . . . . . . 11
β’ β =
βͺ (topGenβran (,)) |
607 | 606 | iuncld 22541 |
. . . . . . . . . 10
β’
(((topGenβran (,)) β Top β§ (1...π) β Fin β§ βπ§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) β βͺ π§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
608 | 1, 599, 605, 607 | mp3an12i 1466 |
. . . . . . . . 9
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ
π§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
609 | 608 | adantr 482 |
. . . . . . . 8
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βͺ π§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran
(,)))) |
610 | | fveq2 6889 |
. . . . . . . . . . . . . . . 16
β’ (π = (πβπ§) β ([,]βπ) = ([,]β(πβπ§))) |
611 | 610 | sseq1d 4013 |
. . . . . . . . . . . . . . 15
β’ (π = (πβπ§) β (([,]βπ) β π΄ β ([,]β(πβπ§)) β π΄)) |
612 | 611 | elrab 3683 |
. . . . . . . . . . . . . 14
β’ ((πβπ§) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β ((πβπ§) β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β§ ([,]β(πβπ§)) β π΄)) |
613 | 612 | simprbi 498 |
. . . . . . . . . . . . 13
β’ ((πβπ§) β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β ([,]β(πβπ§)) β π΄) |
614 | 65, 73, 613 | 3syl 18 |
. . . . . . . . . . . 12
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β β) β ([,]β(πβπ§)) β π΄) |
615 | 556, 614 | sylan2 594 |
. . . . . . . . . . 11
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ π§ β (1...π)) β ([,]β(πβπ§)) β π΄) |
616 | 615 | ralrimiva 3147 |
. . . . . . . . . 10
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βπ§ β (1...π)([,]β(πβπ§)) β π΄) |
617 | | iunss 5048 |
. . . . . . . . . 10
β’ (βͺ π§ β (1...π)([,]β(πβπ§)) β π΄ β βπ§ β (1...π)([,]β(πβπ§)) β π΄) |
618 | 616, 617 | sylibr 233 |
. . . . . . . . 9
β’ (π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β βͺ
π§ β (1...π)([,]β(πβπ§)) β π΄) |
619 | 618 | adantr 482 |
. . . . . . . 8
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βͺ π§ β (1...π)([,]β(πβπ§)) β π΄) |
620 | | simprr 772 |
. . . . . . . 8
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
621 | | sseq1 4007 |
. . . . . . . . . 10
β’ (π = βͺ π§ β (1...π)([,]β(πβπ§)) β (π β π΄ β βͺ
π§ β (1...π)([,]β(πβπ§)) β π΄)) |
622 | | fveq2 6889 |
. . . . . . . . . . 11
β’ (π = βͺ π§ β (1...π)([,]β(πβπ§)) β (vol*βπ ) = (vol*ββͺ π§ β (1...π)([,]β(πβπ§)))) |
623 | 622 | breq2d 5160 |
. . . . . . . . . 10
β’ (π = βͺ π§ β (1...π)([,]β(πβπ§)) β (π < (vol*βπ ) β π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) |
624 | 621, 623 | anbi12d 632 |
. . . . . . . . 9
β’ (π = βͺ π§ β (1...π)([,]β(πβπ§)) β ((π β π΄ β§ π < (vol*βπ )) β (βͺ π§ β (1...π)([,]β(πβπ§)) β π΄ β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§)))))) |
625 | 624 | rspcev 3613 |
. . . . . . . 8
β’
((βͺ π§ β (1...π)([,]β(πβπ§)) β (Clsdβ(topGenβran (,)))
β§ (βͺ π§ β (1...π)([,]β(πβπ§)) β π΄ β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
626 | 609, 619,
620, 625 | syl12anc 836 |
. . . . . . 7
β’ ((π:ββΆ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
627 | 52, 626 | sylan 581 |
. . . . . 6
β’ ((π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)} β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
628 | 627 | adantll 713 |
. . . . 5
β’ ((((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β§ (π β β β§ π < (vol*ββͺ π§ β (1...π)([,]β(πβπ§))))) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
629 | 598, 628 | rexlimddv 3162 |
. . . 4
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
630 | 629 | adantlr 714 |
. . 3
β’ ((((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ β β
) β§ π:ββ1-1-ontoβ{π β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} β£ βπ β {π β ran (π₯ β β€, π¦ β β0 β¦
β¨(π₯ / (2βπ¦)), ((π₯ + 1) / (2βπ¦))β©) β£ ([,]βπ) β π΄} (([,]βπ) β ([,]βπ) β π = π)}) β βπ β (Clsdβ(topGenβran
(,)))(π β π΄ β§ π < (vol*βπ ))) |
631 | 17, 630 | exlimddv 1939 |
. 2
β’ (((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β§ π΄ β β
) β
βπ β
(Clsdβ(topGenβran (,)))(π β π΄ β§ π < (vol*βπ ))) |
632 | 15, 631 | pm2.61dane 3030 |
1
β’ ((π΄ β (topGenβran (,))
β§ π β β
β§ π <
(vol*βπ΄)) β
βπ β
(Clsdβ(topGenβran (,)))(π β π΄ β§ π < (vol*βπ ))) |