Step | Hyp | Ref
| Expression |
1 | | psercn.s |
. . . . . . 7
β’ π = (β‘abs β (0[,)π
)) |
2 | | cnvimass 6037 |
. . . . . . . 8
β’ (β‘abs β (0[,)π
)) β dom abs |
3 | | absf 15231 |
. . . . . . . . 9
β’
abs:ββΆβ |
4 | 3 | fdmi 6684 |
. . . . . . . 8
β’ dom abs =
β |
5 | 2, 4 | sseqtri 3984 |
. . . . . . 7
β’ (β‘abs β (0[,)π
)) β β |
6 | 1, 5 | eqsstri 3982 |
. . . . . 6
β’ π β
β |
7 | 6 | a1i 11 |
. . . . 5
β’ (π β π β β) |
8 | 7 | sselda 3948 |
. . . 4
β’ ((π β§ π β π) β π β β) |
9 | 8 | abscld 15330 |
. . . . 5
β’ ((π β§ π β π) β (absβπ) β β) |
10 | 8 | absge0d 15338 |
. . . . 5
β’ ((π β§ π β π) β 0 β€ (absβπ)) |
11 | | psercnlem2.i |
. . . . . 6
β’ ((π β§ π β π) β (π β β+ β§
(absβπ) < π β§ π < π
)) |
12 | 11 | simp2d 1144 |
. . . . 5
β’ ((π β§ π β π) β (absβπ) < π) |
13 | | 0re 11165 |
. . . . . 6
β’ 0 β
β |
14 | 11 | simp1d 1143 |
. . . . . . 7
β’ ((π β§ π β π) β π β
β+) |
15 | 14 | rpxrd 12966 |
. . . . . 6
β’ ((π β§ π β π) β π β
β*) |
16 | | elico2 13337 |
. . . . . 6
β’ ((0
β β β§ π
β β*) β ((absβπ) β (0[,)π) β ((absβπ) β β β§ 0 β€
(absβπ) β§
(absβπ) < π))) |
17 | 13, 15, 16 | sylancr 588 |
. . . . 5
β’ ((π β§ π β π) β ((absβπ) β (0[,)π) β ((absβπ) β β β§ 0 β€
(absβπ) β§
(absβπ) < π))) |
18 | 9, 10, 12, 17 | mpbir3and 1343 |
. . . 4
β’ ((π β§ π β π) β (absβπ) β (0[,)π)) |
19 | | ffn 6672 |
. . . . 5
β’
(abs:ββΆβ β abs Fn β) |
20 | | elpreima 7012 |
. . . . 5
β’ (abs Fn
β β (π β
(β‘abs β (0[,)π)) β (π β β β§ (absβπ) β (0[,)π)))) |
21 | 3, 19, 20 | mp2b 10 |
. . . 4
β’ (π β (β‘abs β (0[,)π)) β (π β β β§ (absβπ) β (0[,)π))) |
22 | 8, 18, 21 | sylanbrc 584 |
. . 3
β’ ((π β§ π β π) β π β (β‘abs β (0[,)π))) |
23 | | eqid 2733 |
. . . . 5
β’ (abs
β β ) = (abs β β ) |
24 | 23 | cnbl0 24160 |
. . . 4
β’ (π β β*
β (β‘abs β (0[,)π)) = (0(ballβ(abs β
β ))π)) |
25 | 15, 24 | syl 17 |
. . 3
β’ ((π β§ π β π) β (β‘abs β (0[,)π)) = (0(ballβ(abs β β
))π)) |
26 | 22, 25 | eleqtrd 2836 |
. 2
β’ ((π β§ π β π) β π β (0(ballβ(abs β β
))π)) |
27 | | icossicc 13362 |
. . . 4
β’
(0[,)π) β
(0[,]π) |
28 | | imass2 6058 |
. . . 4
β’
((0[,)π) β
(0[,]π) β (β‘abs β (0[,)π)) β (β‘abs β (0[,]π))) |
29 | 27, 28 | mp1i 13 |
. . 3
β’ ((π β§ π β π) β (β‘abs β (0[,)π)) β (β‘abs β (0[,]π))) |
30 | 25, 29 | eqsstrrd 3987 |
. 2
β’ ((π β§ π β π) β (0(ballβ(abs β β
))π) β (β‘abs β (0[,]π))) |
31 | | iccssxr 13356 |
. . . . . 6
β’
(0[,]+β) β β* |
32 | | pserf.g |
. . . . . . . 8
β’ πΊ = (π₯ β β β¦ (π β β0 β¦ ((π΄βπ) Β· (π₯βπ)))) |
33 | | pserf.a |
. . . . . . . 8
β’ (π β π΄:β0βΆβ) |
34 | | pserf.r |
. . . . . . . 8
β’ π
= sup({π β β β£ seq0( + , (πΊβπ)) β dom β }, β*,
< ) |
35 | 32, 33, 34 | radcnvcl 25799 |
. . . . . . 7
β’ (π β π
β (0[,]+β)) |
36 | 35 | adantr 482 |
. . . . . 6
β’ ((π β§ π β π) β π
β (0[,]+β)) |
37 | 31, 36 | sselid 3946 |
. . . . 5
β’ ((π β§ π β π) β π
β
β*) |
38 | 11 | simp3d 1145 |
. . . . 5
β’ ((π β§ π β π) β π < π
) |
39 | | df-ico 13279 |
. . . . . 6
β’ [,) =
(π’ β
β*, π£
β β* β¦ {π€ β β* β£ (π’ β€ π€ β§ π€ < π£)}) |
40 | | df-icc 13280 |
. . . . . 6
β’ [,] =
(π’ β
β*, π£
β β* β¦ {π€ β β* β£ (π’ β€ π€ β§ π€ β€ π£)}) |
41 | | xrlelttr 13084 |
. . . . . 6
β’ ((π§ β β*
β§ π β
β* β§ π
β β*) β ((π§ β€ π β§ π < π
) β π§ < π
)) |
42 | 39, 40, 41 | ixxss2 13292 |
. . . . 5
β’ ((π
β β*
β§ π < π
) β (0[,]π) β (0[,)π
)) |
43 | 37, 38, 42 | syl2anc 585 |
. . . 4
β’ ((π β§ π β π) β (0[,]π) β (0[,)π
)) |
44 | | imass2 6058 |
. . . 4
β’
((0[,]π) β
(0[,)π
) β (β‘abs β (0[,]π)) β (β‘abs β (0[,)π
))) |
45 | 43, 44 | syl 17 |
. . 3
β’ ((π β§ π β π) β (β‘abs β (0[,]π)) β (β‘abs β (0[,)π
))) |
46 | 45, 1 | sseqtrrdi 3999 |
. 2
β’ ((π β§ π β π) β (β‘abs β (0[,]π)) β π) |
47 | 26, 30, 46 | 3jca 1129 |
1
β’ ((π β§ π β π) β (π β (0(ballβ(abs β β
))π) β§
(0(ballβ(abs β β ))π) β (β‘abs β (0[,]π)) β§ (β‘abs β (0[,]π)) β π)) |