Step | Hyp | Ref
| Expression |
1 | | logno1 26007 |
. 2
β’ Β¬
(π₯ β
β+ β¦ (logβπ₯)) β π(1) |
2 | | relogcl 25947 |
. . . . . . 7
β’ (π₯ β β+
β (logβπ₯) β
β) |
3 | 2 | adantl 483 |
. . . . . 6
β’ ((π β§ π₯ β β+) β
(logβπ₯) β
β) |
4 | 3 | recnd 11190 |
. . . . 5
β’ ((π β§ π₯ β β+) β
(logβπ₯) β
β) |
5 | | 2cnd 12238 |
. . . . 5
β’ ((π β§ π₯ β β+) β 2 β
β) |
6 | | 2ne0 12264 |
. . . . . 6
β’ 2 β
0 |
7 | 6 | a1i 11 |
. . . . 5
β’ ((π β§ π₯ β β+) β 2 β
0) |
8 | 4, 5, 7 | divcan2d 11940 |
. . . 4
β’ ((π β§ π₯ β β+) β (2
Β· ((logβπ₯) /
2)) = (logβπ₯)) |
9 | 8 | mpteq2dva 5210 |
. . 3
β’ (π β (π₯ β β+ β¦ (2
Β· ((logβπ₯) /
2))) = (π₯ β
β+ β¦ (logβπ₯))) |
10 | 3 | rehalfcld 12407 |
. . . . 5
β’ ((π β§ π₯ β β+) β
((logβπ₯) / 2) β
β) |
11 | 10 | recnd 11190 |
. . . 4
β’ ((π β§ π₯ β β+) β
((logβπ₯) / 2) β
β) |
12 | | rpssre 12929 |
. . . . . 6
β’
β+ β β |
13 | | 2cn 12235 |
. . . . . 6
β’ 2 β
β |
14 | | o1const 15509 |
. . . . . 6
β’
((β+ β β β§ 2 β β) β
(π₯ β
β+ β¦ 2) β π(1)) |
15 | 12, 13, 14 | mp2an 691 |
. . . . 5
β’ (π₯ β β+
β¦ 2) β π(1) |
16 | 15 | a1i 11 |
. . . 4
β’ (π β (π₯ β β+ β¦ 2) β
π(1)) |
17 | | 1red 11163 |
. . . . 5
β’ (π β 1 β
β) |
18 | | dchrisum0fno1.a |
. . . . 5
β’ (π β (π₯ β β+ β¦
Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ))) β π(1)) |
19 | | sumex 15579 |
. . . . . 6
β’
Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)) β V |
20 | 19 | a1i 11 |
. . . . 5
β’ ((π β§ π₯ β β+) β
Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)) β V) |
21 | 10 | adantrr 716 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
((logβπ₯) / 2) β
β) |
22 | 2 | ad2antrl 727 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (logβπ₯) β
β) |
23 | | log1 25957 |
. . . . . . . . 9
β’
(logβ1) = 0 |
24 | | simprr 772 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 1 β€ π₯) |
25 | | 1rp 12926 |
. . . . . . . . . . 11
β’ 1 β
β+ |
26 | | simprl 770 |
. . . . . . . . . . 11
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π₯ β
β+) |
27 | | logleb 25974 |
. . . . . . . . . . 11
β’ ((1
β β+ β§ π₯ β β+) β (1 β€
π₯ β (logβ1) β€
(logβπ₯))) |
28 | 25, 26, 27 | sylancr 588 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (1 β€ π₯ β (logβ1) β€
(logβπ₯))) |
29 | 24, 28 | mpbid 231 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (logβ1)
β€ (logβπ₯)) |
30 | 23, 29 | eqbrtrrid 5146 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 0 β€
(logβπ₯)) |
31 | | 2re 12234 |
. . . . . . . . 9
β’ 2 β
β |
32 | 31 | a1i 11 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 2 β
β) |
33 | | 2pos 12263 |
. . . . . . . . 9
β’ 0 <
2 |
34 | 33 | a1i 11 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 0 <
2) |
35 | | divge0 12031 |
. . . . . . . 8
β’
((((logβπ₯)
β β β§ 0 β€ (logβπ₯)) β§ (2 β β β§ 0 < 2))
β 0 β€ ((logβπ₯) / 2)) |
36 | 22, 30, 32, 34, 35 | syl22anc 838 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β 0 β€
((logβπ₯) /
2)) |
37 | 21, 36 | absidd 15314 |
. . . . . 6
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβ((logβπ₯) /
2)) = ((logβπ₯) /
2)) |
38 | | fzfid 13885 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(1...(ββπ₯))
β Fin) |
39 | | rpvmasum.z |
. . . . . . . . . . . 12
β’ π =
(β€/nβ€βπ) |
40 | | rpvmasum.l |
. . . . . . . . . . . 12
β’ πΏ = (β€RHomβπ) |
41 | | rpvmasum.a |
. . . . . . . . . . . 12
β’ (π β π β β) |
42 | | rpvmasum2.g |
. . . . . . . . . . . 12
β’ πΊ = (DChrβπ) |
43 | | rpvmasum2.d |
. . . . . . . . . . . 12
β’ π· = (BaseβπΊ) |
44 | | rpvmasum2.1 |
. . . . . . . . . . . 12
β’ 1 =
(0gβπΊ) |
45 | | dchrisum0f.f |
. . . . . . . . . . . 12
β’ πΉ = (π β β β¦ Ξ£π£ β {π β β β£ π β₯ π} (πβ(πΏβπ£))) |
46 | | dchrisum0f.x |
. . . . . . . . . . . 12
β’ (π β π β π·) |
47 | | dchrisum0flb.r |
. . . . . . . . . . . 12
β’ (π β π:(Baseβπ)βΆβ) |
48 | 39, 40, 41, 42, 43, 44, 45, 46, 47 | dchrisum0ff 26871 |
. . . . . . . . . . 11
β’ (π β πΉ:ββΆβ) |
49 | 48 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β πΉ:ββΆβ) |
50 | | elfznn 13477 |
. . . . . . . . . 10
β’ (π β
(1...(ββπ₯))
β π β
β) |
51 | | ffvelcdm 7037 |
. . . . . . . . . 10
β’ ((πΉ:ββΆβ β§
π β β) β
(πΉβπ) β β) |
52 | 49, 50, 51 | syl2an 597 |
. . . . . . . . 9
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (πΉβπ) β
β) |
53 | 50 | adantl 483 |
. . . . . . . . . . 11
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π β
β) |
54 | 53 | nnrpd 12962 |
. . . . . . . . . 10
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π β
β+) |
55 | 54 | rpsqrtcld 15303 |
. . . . . . . . 9
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (ββπ)
β β+) |
56 | 52, 55 | rerpdivcld 12995 |
. . . . . . . 8
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β ((πΉβπ) / (ββπ)) β
β) |
57 | 38, 56 | fsumrecl 15626 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)) β β) |
58 | 57 | recnd 11190 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)) β β) |
59 | 58 | abscld 15328 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβΞ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ))) β β) |
60 | | fzfid 13885 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(1...(ββ(ββπ₯))) β Fin) |
61 | | elfznn 13477 |
. . . . . . . . . . 11
β’ (π β
(1...(ββ(ββπ₯))) β π β β) |
62 | 61 | adantl 483 |
. . . . . . . . . 10
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π β β) |
63 | 62 | nnrecred 12211 |
. . . . . . . . 9
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (1 / π) β β) |
64 | 60, 63 | fsumrecl 15626 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββ(ββπ₯)))(1 / π) β β) |
65 | | logsqrt 26075 |
. . . . . . . . . 10
β’ (π₯ β β+
β (logβ(ββπ₯)) = ((logβπ₯) / 2)) |
66 | 65 | ad2antrl 727 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(logβ(ββπ₯)) = ((logβπ₯) / 2)) |
67 | | rpsqrtcl 15156 |
. . . . . . . . . . 11
β’ (π₯ β β+
β (ββπ₯)
β β+) |
68 | 67 | ad2antrl 727 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(ββπ₯) β
β+) |
69 | | harmoniclbnd 26374 |
. . . . . . . . . 10
β’
((ββπ₯)
β β+ β (logβ(ββπ₯)) β€ Ξ£π β
(1...(ββ(ββπ₯)))(1 / π)) |
70 | 68, 69 | syl 17 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(logβ(ββπ₯)) β€ Ξ£π β
(1...(ββ(ββπ₯)))(1 / π)) |
71 | 66, 70 | eqbrtrrd 5134 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
((logβπ₯) / 2) β€
Ξ£π β
(1...(ββ(ββπ₯)))(1 / π)) |
72 | | eqid 2737 |
. . . . . . . . . . . . . . . . 17
β’ (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) = (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) |
73 | | ovex 7395 |
. . . . . . . . . . . . . . . . 17
β’ (πβ2) β
V |
74 | 72, 73 | elrnmpti 5920 |
. . . . . . . . . . . . . . . 16
β’ (π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) β βπ β
(1...(ββ(ββπ₯)))π = (πβ2)) |
75 | | elfznn 13477 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (π β
(1...(ββ(ββπ₯))) β π β β) |
76 | 75 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π β β) |
77 | 76 | nnrpd 12962 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π β β+) |
78 | 77 | rprege0d 12971 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (π β β β§ 0 β€ π)) |
79 | | sqrtsq 15161 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β β β§ 0 β€
π) β
(ββ(πβ2))
= π) |
80 | 78, 79 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (ββ(πβ2)) = π) |
81 | 80, 76 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (ββ(πβ2)) β
β) |
82 | | fveq2 6847 |
. . . . . . . . . . . . . . . . . . 19
β’ (π = (πβ2) β (ββπ) = (ββ(πβ2))) |
83 | 82 | eleq1d 2823 |
. . . . . . . . . . . . . . . . . 18
β’ (π = (πβ2) β ((ββπ) β β β
(ββ(πβ2))
β β)) |
84 | 81, 83 | syl5ibrcom 247 |
. . . . . . . . . . . . . . . . 17
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (π = (πβ2) β (ββπ) β
β)) |
85 | 84 | rexlimdva 3153 |
. . . . . . . . . . . . . . . 16
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (βπ β
(1...(ββ(ββπ₯)))π = (πβ2) β (ββπ) β
β)) |
86 | 74, 85 | biimtrid 241 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) β (ββπ) β
β)) |
87 | 86 | imp 408 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β (ββπ) β
β) |
88 | 87 | iftrued 4499 |
. . . . . . . . . . . . 13
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β if((ββπ) β β, 1, 0) =
1) |
89 | 88 | oveq1d 7377 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β (if((ββπ) β β, 1, 0) /
(ββπ)) = (1 /
(ββπ))) |
90 | 89 | sumeq2dv 15595 |
. . . . . . . . . . 11
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))(if((ββπ) β β, 1, 0) /
(ββπ)) =
Ξ£π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))(1 / (ββπ))) |
91 | | fveq2 6847 |
. . . . . . . . . . . . 13
β’ (π = (πβ2) β (ββπ) = (ββ(πβ2))) |
92 | 91 | oveq2d 7378 |
. . . . . . . . . . . 12
β’ (π = (πβ2) β (1 / (ββπ)) = (1 / (ββ(πβ2)))) |
93 | 76 | nnsqcld 14154 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (πβ2) β β) |
94 | 68 | rpred 12964 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(ββπ₯) β
β) |
95 | | fznnfl 13774 |
. . . . . . . . . . . . . . . . . . . 20
β’
((ββπ₯)
β β β (π
β (1...(ββ(ββπ₯))) β (π β β β§ π β€ (ββπ₯)))) |
96 | 94, 95 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β
(1...(ββ(ββπ₯))) β (π β β β§ π β€ (ββπ₯)))) |
97 | 96 | simplbda 501 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π β€ (ββπ₯)) |
98 | 68 | adantr 482 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (ββπ₯) β
β+) |
99 | 98 | rprege0d 12971 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β ((ββπ₯) β β β§ 0 β€
(ββπ₯))) |
100 | | le2sq 14046 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β β β§ 0 β€
π) β§
((ββπ₯) β
β β§ 0 β€ (ββπ₯))) β (π β€ (ββπ₯) β (πβ2) β€ ((ββπ₯)β2))) |
101 | 78, 99, 100 | syl2anc 585 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (π β€ (ββπ₯) β (πβ2) β€ ((ββπ₯)β2))) |
102 | 97, 101 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (πβ2) β€ ((ββπ₯)β2)) |
103 | 26 | rpred 12964 |
. . . . . . . . . . . . . . . . . . . 20
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β π₯ β
β) |
104 | 103 | adantr 482 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π₯ β β) |
105 | 104 | recnd 11190 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π₯ β β) |
106 | 105 | sqsqrtd 15331 |
. . . . . . . . . . . . . . . . 17
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β ((ββπ₯)β2) = π₯) |
107 | 102, 106 | breqtrd 5136 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (πβ2) β€ π₯) |
108 | | fznnfl 13774 |
. . . . . . . . . . . . . . . . 17
β’ (π₯ β β β ((πβ2) β
(1...(ββπ₯))
β ((πβ2) β
β β§ (πβ2)
β€ π₯))) |
109 | 104, 108 | syl 17 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β ((πβ2) β (1...(ββπ₯)) β ((πβ2) β β β§ (πβ2) β€ π₯))) |
110 | 93, 107, 109 | mpbir2and 712 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (πβ2) β (1...(ββπ₯))) |
111 | 110 | ex 414 |
. . . . . . . . . . . . . 14
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β
(1...(ββ(ββπ₯))) β (πβ2) β (1...(ββπ₯)))) |
112 | 75 | nnrpd 12962 |
. . . . . . . . . . . . . . . . 17
β’ (π β
(1...(ββ(ββπ₯))) β π β β+) |
113 | 112 | rprege0d 12971 |
. . . . . . . . . . . . . . . 16
β’ (π β
(1...(ββ(ββπ₯))) β (π β β β§ 0 β€ π)) |
114 | 61 | nnrpd 12962 |
. . . . . . . . . . . . . . . . 17
β’ (π β
(1...(ββ(ββπ₯))) β π β β+) |
115 | 114 | rprege0d 12971 |
. . . . . . . . . . . . . . . 16
β’ (π β
(1...(ββ(ββπ₯))) β (π β β β§ 0 β€ π)) |
116 | | sq11 14043 |
. . . . . . . . . . . . . . . 16
β’ (((π β β β§ 0 β€
π) β§ (π β β β§ 0 β€
π)) β ((πβ2) = (πβ2) β π = π)) |
117 | 113, 115,
116 | syl2an 597 |
. . . . . . . . . . . . . . 15
β’ ((π β
(1...(ββ(ββπ₯))) β§ π β
(1...(ββ(ββπ₯)))) β ((πβ2) = (πβ2) β π = π)) |
118 | 117 | a1i 11 |
. . . . . . . . . . . . . 14
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β ((π β
(1...(ββ(ββπ₯))) β§ π β
(1...(ββ(ββπ₯)))) β ((πβ2) = (πβ2) β π = π))) |
119 | 111, 118 | dom2lem 8939 |
. . . . . . . . . . . . 13
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))β1-1β(1...(ββπ₯))) |
120 | | f1f1orn 6800 |
. . . . . . . . . . . . 13
β’ ((π β
(1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))β1-1β(1...(ββπ₯)) β (π β (1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))β1-1-ontoβran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) |
121 | 119, 120 | syl 17 |
. . . . . . . . . . . 12
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))β1-1-ontoβran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) |
122 | | oveq1 7369 |
. . . . . . . . . . . . . 14
β’ (π = π β (πβ2) = (πβ2)) |
123 | 122, 72, 73 | fvmpt3i 6958 |
. . . . . . . . . . . . 13
β’ (π β
(1...(ββ(ββπ₯))) β ((π β
(1...(ββ(ββπ₯))) β¦ (πβ2))βπ) = (πβ2)) |
124 | 123 | adantl 483 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β ((π β
(1...(ββ(ββπ₯))) β¦ (πβ2))βπ) = (πβ2)) |
125 | | f1f 6743 |
. . . . . . . . . . . . . . . 16
β’ ((π β
(1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))β1-1β(1...(ββπ₯)) β (π β (1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))βΆ(1...(ββπ₯))) |
126 | | frn 6680 |
. . . . . . . . . . . . . . . 16
β’ ((π β
(1...(ββ(ββπ₯))) β¦ (πβ2)):(1...(ββ(ββπ₯)))βΆ(1...(ββπ₯)) β ran (π β (1...(ββ(ββπ₯))) β¦ (πβ2)) β (1...(ββπ₯))) |
127 | 119, 125,
126 | 3syl 18 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) β (1...(ββπ₯))) |
128 | 127 | sselda 3949 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β π β (1...(ββπ₯))) |
129 | | 1re 11162 |
. . . . . . . . . . . . . . . . 17
β’ 1 β
β |
130 | | 0re 11164 |
. . . . . . . . . . . . . . . . 17
β’ 0 β
β |
131 | 129, 130 | ifcli 4538 |
. . . . . . . . . . . . . . . 16
β’
if((ββπ)
β β, 1, 0) β β |
132 | | rerpdivcl 12952 |
. . . . . . . . . . . . . . . 16
β’
((if((ββπ) β β, 1, 0) β β β§
(ββπ) β
β+) β (if((ββπ) β β, 1, 0) /
(ββπ)) β
β) |
133 | 131, 55, 132 | sylancr 588 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (if((ββπ) β β, 1, 0) /
(ββπ)) β
β) |
134 | 133 | recnd 11190 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (if((ββπ) β β, 1, 0) /
(ββπ)) β
β) |
135 | 128, 134 | syldan 592 |
. . . . . . . . . . . . 13
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β (if((ββπ) β β, 1, 0) /
(ββπ)) β
β) |
136 | 89, 135 | eqeltrrd 2839 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β (1 / (ββπ)) β
β) |
137 | 92, 60, 121, 124, 136 | fsumf1o 15615 |
. . . . . . . . . . 11
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))(1 / (ββπ)) = Ξ£π β
(1...(ββ(ββπ₯)))(1 / (ββ(πβ2)))) |
138 | 90, 137 | eqtrd 2777 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))(if((ββπ) β β, 1, 0) /
(ββπ)) =
Ξ£π β
(1...(ββ(ββπ₯)))(1 / (ββ(πβ2)))) |
139 | | eldif 3925 |
. . . . . . . . . . . . . . 15
β’ (π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β (π β (1...(ββπ₯)) β§ Β¬ π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) |
140 | 50 | ad2antrl 727 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π
β β) |
141 | 140 | nncnd 12176 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π
β β) |
142 | 141 | sqsqrtd 15331 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β ((ββπ)β2) = π) |
143 | | simprr 772 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (ββπ) β β) |
144 | | fznnfl 13774 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
β’ (π₯ β β β (π β
(1...(ββπ₯))
β (π β β
β§ π β€ π₯))) |
145 | 103, 144 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β (π β
(1...(ββπ₯))
β (π β β
β§ π β€ π₯))) |
146 | 145 | simplbda 501 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π β€ π₯) |
147 | 146 | adantrr 716 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π
β€ π₯) |
148 | 140 | nnrpd 12962 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π
β β+) |
149 | 148 | rprege0d 12971 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (π
β β β§ 0 β€ π)) |
150 | 26 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π₯
β β+) |
151 | 150 | rprege0d 12971 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (π₯
β β β§ 0 β€ π₯)) |
152 | | sqrtle 15152 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β β β§ 0 β€
π) β§ (π₯ β β β§ 0 β€
π₯)) β (π β€ π₯ β (ββπ) β€ (ββπ₯))) |
153 | 149, 151,
152 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (π
β€ π₯ β
(ββπ) β€
(ββπ₯))) |
154 | 147, 153 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (ββπ) β€ (ββπ₯)) |
155 | 68 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (ββπ₯) β
β+) |
156 | 155 | rpred 12964 |
. . . . . . . . . . . . . . . . . . . . . 22
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (ββπ₯) β β) |
157 | | fznnfl 13774 |
. . . . . . . . . . . . . . . . . . . . . 22
β’
((ββπ₯)
β β β ((ββπ) β
(1...(ββ(ββπ₯))) β ((ββπ) β β β§ (ββπ) β€ (ββπ₯)))) |
158 | 156, 157 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β ((ββπ) β
(1...(ββ(ββπ₯))) β ((ββπ) β β β§ (ββπ) β€ (ββπ₯)))) |
159 | 143, 154,
158 | mpbir2and 712 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β (ββπ) β
(1...(ββ(ββπ₯)))) |
160 | 142, 140 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . . . 20
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β ((ββπ)β2) β β) |
161 | | oveq1 7369 |
. . . . . . . . . . . . . . . . . . . . 21
β’ (π = (ββπ) β (πβ2) = ((ββπ)β2)) |
162 | 72, 161 | elrnmpt1s 5917 |
. . . . . . . . . . . . . . . . . . . 20
β’
(((ββπ)
β (1...(ββ(ββπ₯))) β§ ((ββπ)β2) β β) β
((ββπ)β2)
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) |
163 | 159, 160,
162 | syl2anc 585 |
. . . . . . . . . . . . . . . . . . 19
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β ((ββπ)β2) β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) |
164 | 142, 163 | eqeltrrd 2839 |
. . . . . . . . . . . . . . . . . 18
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ (ββπ)
β β)) β π
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) |
165 | 164 | expr 458 |
. . . . . . . . . . . . . . . . 17
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β ((ββπ)
β β β π
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) |
166 | 165 | con3d 152 |
. . . . . . . . . . . . . . . 16
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (Β¬ π β ran
(π β
(1...(ββ(ββπ₯))) β¦ (πβ2)) β Β¬ (ββπ) β
β)) |
167 | 166 | impr 456 |
. . . . . . . . . . . . . . 15
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ (π β
(1...(ββπ₯))
β§ Β¬ π β ran
(π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β Β¬ (ββπ) β
β) |
168 | 139, 167 | sylan2b 595 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β Β¬ (ββπ) β
β) |
169 | 168 | iffalsed 4502 |
. . . . . . . . . . . . 13
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β if((ββπ) β β, 1, 0) =
0) |
170 | 169 | oveq1d 7377 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β (if((ββπ) β β, 1, 0) /
(ββπ)) = (0 /
(ββπ))) |
171 | | eldifi 4091 |
. . . . . . . . . . . . . . 15
β’ (π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))) β π β (1...(ββπ₯))) |
172 | 171, 55 | sylan2 594 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β (ββπ) β
β+) |
173 | 172 | rpcnne0d 12973 |
. . . . . . . . . . . . 13
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β ((ββπ) β β β§
(ββπ) β
0)) |
174 | | div0 11850 |
. . . . . . . . . . . . 13
β’
(((ββπ)
β β β§ (ββπ) β 0) β (0 / (ββπ)) = 0) |
175 | 173, 174 | syl 17 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β (0 / (ββπ)) = 0) |
176 | 170, 175 | eqtrd 2777 |
. . . . . . . . . . 11
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
((1...(ββπ₯))
β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2)))) β (if((ββπ) β β, 1, 0) /
(ββπ)) =
0) |
177 | 127, 135,
176, 38 | fsumss 15617 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β ran (π β
(1...(ββ(ββπ₯))) β¦ (πβ2))(if((ββπ) β β, 1, 0) /
(ββπ)) =
Ξ£π β
(1...(ββπ₯))(if((ββπ) β β, 1, 0) /
(ββπ))) |
178 | 62 | nnrpd 12962 |
. . . . . . . . . . . . . 14
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β π β β+) |
179 | 178 | rprege0d 12971 |
. . . . . . . . . . . . 13
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (π β β β§ 0 β€ π)) |
180 | | sqrtsq 15161 |
. . . . . . . . . . . . 13
β’ ((π β β β§ 0 β€
π) β
(ββ(πβ2))
= π) |
181 | 179, 180 | syl 17 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (ββ(πβ2)) = π) |
182 | 181 | oveq2d 7378 |
. . . . . . . . . . 11
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββ(ββπ₯)))) β (1 / (ββ(πβ2))) = (1 / π)) |
183 | 182 | sumeq2dv 15595 |
. . . . . . . . . 10
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββ(ββπ₯)))(1 / (ββ(πβ2))) = Ξ£π β
(1...(ββ(ββπ₯)))(1 / π)) |
184 | 138, 177,
183 | 3eqtr3d 2785 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββπ₯))(if((ββπ) β β, 1, 0) /
(ββπ)) =
Ξ£π β
(1...(ββ(ββπ₯)))(1 / π)) |
185 | 131 | a1i 11 |
. . . . . . . . . . 11
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β if((ββπ)
β β, 1, 0) β β) |
186 | 41 | ad2antrr 725 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π β
β) |
187 | 46 | ad2antrr 725 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π β π·) |
188 | 47 | ad2antrr 725 |
. . . . . . . . . . . 12
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β π:(Baseβπ)βΆβ) |
189 | 39, 40, 186, 42, 43, 44, 45, 187, 188, 53 | dchrisum0flb 26874 |
. . . . . . . . . . 11
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β if((ββπ)
β β, 1, 0) β€ (πΉβπ)) |
190 | 185, 52, 55, 189 | lediv1dd 13022 |
. . . . . . . . . 10
β’ (((π β§ (π₯ β β+ β§ 1 β€
π₯)) β§ π β
(1...(ββπ₯)))
β (if((ββπ) β β, 1, 0) /
(ββπ)) β€
((πΉβπ) / (ββπ))) |
191 | 38, 133, 56, 190 | fsumle 15691 |
. . . . . . . . 9
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββπ₯))(if((ββπ) β β, 1, 0) /
(ββπ)) β€
Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ))) |
192 | 184, 191 | eqbrtrrd 5134 |
. . . . . . . 8
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββ(ββπ₯)))(1 / π) β€ Ξ£π β (1...(ββπ₯))((πΉβπ) / (ββπ))) |
193 | 21, 64, 57, 71, 192 | letrd 11319 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
((logβπ₯) / 2) β€
Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ))) |
194 | 57 | leabsd 15306 |
. . . . . . 7
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β Ξ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)) β€ (absβΞ£π β (1...(ββπ₯))((πΉβπ) / (ββπ)))) |
195 | 21, 57, 59, 193, 194 | letrd 11319 |
. . . . . 6
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
((logβπ₯) / 2) β€
(absβΞ£π β
(1...(ββπ₯))((πΉβπ) / (ββπ)))) |
196 | 37, 195 | eqbrtrd 5132 |
. . . . 5
β’ ((π β§ (π₯ β β+ β§ 1 β€
π₯)) β
(absβ((logβπ₯) /
2)) β€ (absβΞ£π β (1...(ββπ₯))((πΉβπ) / (ββπ)))) |
197 | 17, 18, 20, 11, 196 | o1le 15544 |
. . . 4
β’ (π β (π₯ β β+ β¦
((logβπ₯) / 2)) β
π(1)) |
198 | 5, 11, 16, 197 | o1mul2 15514 |
. . 3
β’ (π β (π₯ β β+ β¦ (2
Β· ((logβπ₯) /
2))) β π(1)) |
199 | 9, 198 | eqeltrrd 2839 |
. 2
β’ (π β (π₯ β β+ β¦
(logβπ₯)) β
π(1)) |
200 | 1, 199 | mto 196 |
1
β’ Β¬
π |