Step |
Hyp |
Ref |
Expression |
1 |
|
fzfid |
|- ( A e. NN -> ( 2 ... A ) e. Fin ) |
2 |
|
elfzuz |
|- ( n e. ( 2 ... A ) -> n e. ( ZZ>= ` 2 ) ) |
3 |
|
eluz2nn |
|- ( n e. ( ZZ>= ` 2 ) -> n e. NN ) |
4 |
2 3
|
syl |
|- ( n e. ( 2 ... A ) -> n e. NN ) |
5 |
4
|
adantl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> n e. NN ) |
6 |
5
|
nnrpd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> n e. RR+ ) |
7 |
6
|
relogcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( log ` n ) e. RR ) |
8 |
2
|
adantl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> n e. ( ZZ>= ` 2 ) ) |
9 |
|
uz2m1nn |
|- ( n e. ( ZZ>= ` 2 ) -> ( n - 1 ) e. NN ) |
10 |
8 9
|
syl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - 1 ) e. NN ) |
11 |
5 10
|
nnmulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n x. ( n - 1 ) ) e. NN ) |
12 |
7 11
|
nndivred |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( log ` n ) / ( n x. ( n - 1 ) ) ) e. RR ) |
13 |
1 12
|
fsumrecl |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( log ` n ) / ( n x. ( n - 1 ) ) ) e. RR ) |
14 |
|
2re |
|- 2 e. RR |
15 |
10
|
nnrpd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - 1 ) e. RR+ ) |
16 |
15
|
rpsqrtcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) e. RR+ ) |
17 |
|
rerpdivcl |
|- ( ( 2 e. RR /\ ( sqrt ` ( n - 1 ) ) e. RR+ ) -> ( 2 / ( sqrt ` ( n - 1 ) ) ) e. RR ) |
18 |
14 16 17
|
sylancr |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 / ( sqrt ` ( n - 1 ) ) ) e. RR ) |
19 |
6
|
rpsqrtcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` n ) e. RR+ ) |
20 |
|
rerpdivcl |
|- ( ( 2 e. RR /\ ( sqrt ` n ) e. RR+ ) -> ( 2 / ( sqrt ` n ) ) e. RR ) |
21 |
14 19 20
|
sylancr |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 / ( sqrt ` n ) ) e. RR ) |
22 |
18 21
|
resubcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) e. RR ) |
23 |
1 22
|
fsumrecl |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) e. RR ) |
24 |
14
|
a1i |
|- ( A e. NN -> 2 e. RR ) |
25 |
16
|
rpred |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) e. RR ) |
26 |
5
|
nnred |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> n e. RR ) |
27 |
|
peano2rem |
|- ( n e. RR -> ( n - 1 ) e. RR ) |
28 |
26 27
|
syl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - 1 ) e. RR ) |
29 |
26 28
|
remulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n x. ( n - 1 ) ) e. RR ) |
30 |
29 22
|
remulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) e. RR ) |
31 |
5
|
nncnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> n e. CC ) |
32 |
|
ax-1cn |
|- 1 e. CC |
33 |
|
npcan |
|- ( ( n e. CC /\ 1 e. CC ) -> ( ( n - 1 ) + 1 ) = n ) |
34 |
31 32 33
|
sylancl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n - 1 ) + 1 ) = n ) |
35 |
34
|
fveq2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( log ` ( ( n - 1 ) + 1 ) ) = ( log ` n ) ) |
36 |
15
|
rpge0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 0 <_ ( n - 1 ) ) |
37 |
|
loglesqrt |
|- ( ( ( n - 1 ) e. RR /\ 0 <_ ( n - 1 ) ) -> ( log ` ( ( n - 1 ) + 1 ) ) <_ ( sqrt ` ( n - 1 ) ) ) |
38 |
28 36 37
|
syl2anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( log ` ( ( n - 1 ) + 1 ) ) <_ ( sqrt ` ( n - 1 ) ) ) |
39 |
35 38
|
eqbrtrrd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( log ` n ) <_ ( sqrt ` ( n - 1 ) ) ) |
40 |
19
|
rpred |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` n ) e. RR ) |
41 |
40 25
|
readdcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) e. RR ) |
42 |
|
remulcl |
|- ( ( ( sqrt ` n ) e. RR /\ 2 e. RR ) -> ( ( sqrt ` n ) x. 2 ) e. RR ) |
43 |
40 14 42
|
sylancl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. 2 ) e. RR ) |
44 |
40 25
|
resubcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) e. RR ) |
45 |
26
|
lem1d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - 1 ) <_ n ) |
46 |
6
|
rpge0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 0 <_ n ) |
47 |
28 36 26 46
|
sqrtled |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n - 1 ) <_ n <-> ( sqrt ` ( n - 1 ) ) <_ ( sqrt ` n ) ) ) |
48 |
45 47
|
mpbid |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) <_ ( sqrt ` n ) ) |
49 |
40 25
|
subge0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 0 <_ ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) <-> ( sqrt ` ( n - 1 ) ) <_ ( sqrt ` n ) ) ) |
50 |
48 49
|
mpbird |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 0 <_ ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) |
51 |
25 40 40 48
|
leadd2dd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) <_ ( ( sqrt ` n ) + ( sqrt ` n ) ) ) |
52 |
19
|
rpcnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` n ) e. CC ) |
53 |
52
|
times2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. 2 ) = ( ( sqrt ` n ) + ( sqrt ` n ) ) ) |
54 |
51 53
|
breqtrrd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) <_ ( ( sqrt ` n ) x. 2 ) ) |
55 |
41 43 44 50 54
|
lemul1ad |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) <_ ( ( ( sqrt ` n ) x. 2 ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) |
56 |
31
|
sqsqrtd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) ^ 2 ) = n ) |
57 |
|
subcl |
|- ( ( n e. CC /\ 1 e. CC ) -> ( n - 1 ) e. CC ) |
58 |
31 32 57
|
sylancl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - 1 ) e. CC ) |
59 |
58
|
sqsqrtd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` ( n - 1 ) ) ^ 2 ) = ( n - 1 ) ) |
60 |
56 59
|
oveq12d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) ^ 2 ) - ( ( sqrt ` ( n - 1 ) ) ^ 2 ) ) = ( n - ( n - 1 ) ) ) |
61 |
16
|
rpcnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) e. CC ) |
62 |
|
subsq |
|- ( ( ( sqrt ` n ) e. CC /\ ( sqrt ` ( n - 1 ) ) e. CC ) -> ( ( ( sqrt ` n ) ^ 2 ) - ( ( sqrt ` ( n - 1 ) ) ^ 2 ) ) = ( ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) |
63 |
52 61 62
|
syl2anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) ^ 2 ) - ( ( sqrt ` ( n - 1 ) ) ^ 2 ) ) = ( ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) |
64 |
|
nncan |
|- ( ( n e. CC /\ 1 e. CC ) -> ( n - ( n - 1 ) ) = 1 ) |
65 |
31 32 64
|
sylancl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n - ( n - 1 ) ) = 1 ) |
66 |
60 63 65
|
3eqtr3d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) + ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) = 1 ) |
67 |
|
2cn |
|- 2 e. CC |
68 |
67
|
a1i |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 2 e. CC ) |
69 |
44
|
recnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) e. CC ) |
70 |
52 68 69
|
mulassd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. 2 ) x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) = ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
71 |
55 66 70
|
3brtr3d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 1 <_ ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
72 |
|
1red |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 1 e. RR ) |
73 |
|
remulcl |
|- ( ( 2 e. RR /\ ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) e. RR ) -> ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) e. RR ) |
74 |
14 44 73
|
sylancr |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) e. RR ) |
75 |
40 74
|
remulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) e. RR ) |
76 |
72 75 16
|
lemul1d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 1 <_ ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) <-> ( 1 x. ( sqrt ` ( n - 1 ) ) ) <_ ( ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) x. ( sqrt ` ( n - 1 ) ) ) ) ) |
77 |
71 76
|
mpbid |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 1 x. ( sqrt ` ( n - 1 ) ) ) <_ ( ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) x. ( sqrt ` ( n - 1 ) ) ) ) |
78 |
61
|
mulid2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 1 x. ( sqrt ` ( n - 1 ) ) ) = ( sqrt ` ( n - 1 ) ) ) |
79 |
74
|
recnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) e. CC ) |
80 |
52 79 61
|
mul32d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) x. ( sqrt ` ( n - 1 ) ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
81 |
77 78 80
|
3brtr3d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) <_ ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
82 |
|
remsqsqrt |
|- ( ( n e. RR /\ 0 <_ n ) -> ( ( sqrt ` n ) x. ( sqrt ` n ) ) = n ) |
83 |
26 46 82
|
syl2anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( sqrt ` n ) ) = n ) |
84 |
|
remsqsqrt |
|- ( ( ( n - 1 ) e. RR /\ 0 <_ ( n - 1 ) ) -> ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` ( n - 1 ) ) ) = ( n - 1 ) ) |
85 |
28 36 84
|
syl2anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` ( n - 1 ) ) ) = ( n - 1 ) ) |
86 |
83 85
|
oveq12d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. ( sqrt ` n ) ) x. ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` ( n - 1 ) ) ) ) = ( n x. ( n - 1 ) ) ) |
87 |
52 52 61 61
|
mul4d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. ( sqrt ` n ) ) x. ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` ( n - 1 ) ) ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) |
88 |
86 87
|
eqtr3d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( n x. ( n - 1 ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) |
89 |
16
|
rpcnne0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` ( n - 1 ) ) e. CC /\ ( sqrt ` ( n - 1 ) ) =/= 0 ) ) |
90 |
19
|
rpcnne0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) e. CC /\ ( sqrt ` n ) =/= 0 ) ) |
91 |
|
divsubdiv |
|- ( ( ( 2 e. CC /\ 2 e. CC ) /\ ( ( ( sqrt ` ( n - 1 ) ) e. CC /\ ( sqrt ` ( n - 1 ) ) =/= 0 ) /\ ( ( sqrt ` n ) e. CC /\ ( sqrt ` n ) =/= 0 ) ) ) -> ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) = ( ( ( 2 x. ( sqrt ` n ) ) - ( 2 x. ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` n ) ) ) ) |
92 |
68 68 89 90 91
|
syl22anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) = ( ( ( 2 x. ( sqrt ` n ) ) - ( 2 x. ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` n ) ) ) ) |
93 |
68 52 61
|
subdid |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) = ( ( 2 x. ( sqrt ` n ) ) - ( 2 x. ( sqrt ` ( n - 1 ) ) ) ) ) |
94 |
52 61
|
mulcomd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) = ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` n ) ) ) |
95 |
93 94
|
oveq12d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) = ( ( ( 2 x. ( sqrt ` n ) ) - ( 2 x. ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` ( n - 1 ) ) x. ( sqrt ` n ) ) ) ) |
96 |
92 95
|
eqtr4d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) = ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) |
97 |
88 96
|
oveq12d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) = ( ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) x. ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) ) |
98 |
52 61
|
mulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) e. CC ) |
99 |
19 16
|
rpmulcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) e. RR+ ) |
100 |
74 99
|
rerpdivcld |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) e. RR ) |
101 |
100
|
recnd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) e. CC ) |
102 |
98 98 101
|
mulassd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) x. ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) ) ) |
103 |
99
|
rpne0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) =/= 0 ) |
104 |
79 98 103
|
divcan2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) = ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) |
105 |
104
|
oveq2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) / ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) ) ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
106 |
97 102 105
|
3eqtrd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) = ( ( ( sqrt ` n ) x. ( sqrt ` ( n - 1 ) ) ) x. ( 2 x. ( ( sqrt ` n ) - ( sqrt ` ( n - 1 ) ) ) ) ) ) |
107 |
81 106
|
breqtrrd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( n - 1 ) ) <_ ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) ) |
108 |
7 25 30 39 107
|
letrd |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( log ` n ) <_ ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) ) |
109 |
11
|
nngt0d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> 0 < ( n x. ( n - 1 ) ) ) |
110 |
|
ledivmul |
|- ( ( ( log ` n ) e. RR /\ ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) e. RR /\ ( ( n x. ( n - 1 ) ) e. RR /\ 0 < ( n x. ( n - 1 ) ) ) ) -> ( ( ( log ` n ) / ( n x. ( n - 1 ) ) ) <_ ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) <-> ( log ` n ) <_ ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) ) ) |
111 |
7 22 29 109 110
|
syl112anc |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( ( log ` n ) / ( n x. ( n - 1 ) ) ) <_ ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) <-> ( log ` n ) <_ ( ( n x. ( n - 1 ) ) x. ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) ) ) |
112 |
108 111
|
mpbird |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( log ` n ) / ( n x. ( n - 1 ) ) ) <_ ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) |
113 |
1 12 22 112
|
fsumle |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( log ` n ) / ( n x. ( n - 1 ) ) ) <_ sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) |
114 |
|
fvoveq1 |
|- ( k = n -> ( sqrt ` ( k - 1 ) ) = ( sqrt ` ( n - 1 ) ) ) |
115 |
114
|
oveq2d |
|- ( k = n -> ( 2 / ( sqrt ` ( k - 1 ) ) ) = ( 2 / ( sqrt ` ( n - 1 ) ) ) ) |
116 |
|
fvoveq1 |
|- ( k = ( n + 1 ) -> ( sqrt ` ( k - 1 ) ) = ( sqrt ` ( ( n + 1 ) - 1 ) ) ) |
117 |
116
|
oveq2d |
|- ( k = ( n + 1 ) -> ( 2 / ( sqrt ` ( k - 1 ) ) ) = ( 2 / ( sqrt ` ( ( n + 1 ) - 1 ) ) ) ) |
118 |
|
oveq1 |
|- ( k = 2 -> ( k - 1 ) = ( 2 - 1 ) ) |
119 |
|
2m1e1 |
|- ( 2 - 1 ) = 1 |
120 |
118 119
|
eqtrdi |
|- ( k = 2 -> ( k - 1 ) = 1 ) |
121 |
120
|
fveq2d |
|- ( k = 2 -> ( sqrt ` ( k - 1 ) ) = ( sqrt ` 1 ) ) |
122 |
|
sqrt1 |
|- ( sqrt ` 1 ) = 1 |
123 |
121 122
|
eqtrdi |
|- ( k = 2 -> ( sqrt ` ( k - 1 ) ) = 1 ) |
124 |
123
|
oveq2d |
|- ( k = 2 -> ( 2 / ( sqrt ` ( k - 1 ) ) ) = ( 2 / 1 ) ) |
125 |
67
|
div1i |
|- ( 2 / 1 ) = 2 |
126 |
124 125
|
eqtrdi |
|- ( k = 2 -> ( 2 / ( sqrt ` ( k - 1 ) ) ) = 2 ) |
127 |
|
fvoveq1 |
|- ( k = ( A + 1 ) -> ( sqrt ` ( k - 1 ) ) = ( sqrt ` ( ( A + 1 ) - 1 ) ) ) |
128 |
127
|
oveq2d |
|- ( k = ( A + 1 ) -> ( 2 / ( sqrt ` ( k - 1 ) ) ) = ( 2 / ( sqrt ` ( ( A + 1 ) - 1 ) ) ) ) |
129 |
|
nnz |
|- ( A e. NN -> A e. ZZ ) |
130 |
|
eluzp1p1 |
|- ( A e. ( ZZ>= ` 1 ) -> ( A + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) ) |
131 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
132 |
130 131
|
eleq2s |
|- ( A e. NN -> ( A + 1 ) e. ( ZZ>= ` ( 1 + 1 ) ) ) |
133 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
134 |
133
|
fveq2i |
|- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) ) |
135 |
132 134
|
eleqtrrdi |
|- ( A e. NN -> ( A + 1 ) e. ( ZZ>= ` 2 ) ) |
136 |
|
elfzuz |
|- ( k e. ( 2 ... ( A + 1 ) ) -> k e. ( ZZ>= ` 2 ) ) |
137 |
|
uz2m1nn |
|- ( k e. ( ZZ>= ` 2 ) -> ( k - 1 ) e. NN ) |
138 |
136 137
|
syl |
|- ( k e. ( 2 ... ( A + 1 ) ) -> ( k - 1 ) e. NN ) |
139 |
138
|
adantl |
|- ( ( A e. NN /\ k e. ( 2 ... ( A + 1 ) ) ) -> ( k - 1 ) e. NN ) |
140 |
139
|
nnrpd |
|- ( ( A e. NN /\ k e. ( 2 ... ( A + 1 ) ) ) -> ( k - 1 ) e. RR+ ) |
141 |
140
|
rpsqrtcld |
|- ( ( A e. NN /\ k e. ( 2 ... ( A + 1 ) ) ) -> ( sqrt ` ( k - 1 ) ) e. RR+ ) |
142 |
|
rerpdivcl |
|- ( ( 2 e. RR /\ ( sqrt ` ( k - 1 ) ) e. RR+ ) -> ( 2 / ( sqrt ` ( k - 1 ) ) ) e. RR ) |
143 |
14 141 142
|
sylancr |
|- ( ( A e. NN /\ k e. ( 2 ... ( A + 1 ) ) ) -> ( 2 / ( sqrt ` ( k - 1 ) ) ) e. RR ) |
144 |
143
|
recnd |
|- ( ( A e. NN /\ k e. ( 2 ... ( A + 1 ) ) ) -> ( 2 / ( sqrt ` ( k - 1 ) ) ) e. CC ) |
145 |
115 117 126 128 129 135 144
|
telfsum |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` ( ( n + 1 ) - 1 ) ) ) ) = ( 2 - ( 2 / ( sqrt ` ( ( A + 1 ) - 1 ) ) ) ) ) |
146 |
|
pncan |
|- ( ( n e. CC /\ 1 e. CC ) -> ( ( n + 1 ) - 1 ) = n ) |
147 |
31 32 146
|
sylancl |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( n + 1 ) - 1 ) = n ) |
148 |
147
|
fveq2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( sqrt ` ( ( n + 1 ) - 1 ) ) = ( sqrt ` n ) ) |
149 |
148
|
oveq2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( 2 / ( sqrt ` ( ( n + 1 ) - 1 ) ) ) = ( 2 / ( sqrt ` n ) ) ) |
150 |
149
|
oveq2d |
|- ( ( A e. NN /\ n e. ( 2 ... A ) ) -> ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` ( ( n + 1 ) - 1 ) ) ) ) = ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) |
151 |
150
|
sumeq2dv |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` ( ( n + 1 ) - 1 ) ) ) ) = sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) ) |
152 |
|
nncn |
|- ( A e. NN -> A e. CC ) |
153 |
|
pncan |
|- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A ) |
154 |
152 32 153
|
sylancl |
|- ( A e. NN -> ( ( A + 1 ) - 1 ) = A ) |
155 |
154
|
fveq2d |
|- ( A e. NN -> ( sqrt ` ( ( A + 1 ) - 1 ) ) = ( sqrt ` A ) ) |
156 |
155
|
oveq2d |
|- ( A e. NN -> ( 2 / ( sqrt ` ( ( A + 1 ) - 1 ) ) ) = ( 2 / ( sqrt ` A ) ) ) |
157 |
156
|
oveq2d |
|- ( A e. NN -> ( 2 - ( 2 / ( sqrt ` ( ( A + 1 ) - 1 ) ) ) ) = ( 2 - ( 2 / ( sqrt ` A ) ) ) ) |
158 |
145 151 157
|
3eqtr3d |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) = ( 2 - ( 2 / ( sqrt ` A ) ) ) ) |
159 |
|
2rp |
|- 2 e. RR+ |
160 |
|
nnrp |
|- ( A e. NN -> A e. RR+ ) |
161 |
160
|
rpsqrtcld |
|- ( A e. NN -> ( sqrt ` A ) e. RR+ ) |
162 |
|
rpdivcl |
|- ( ( 2 e. RR+ /\ ( sqrt ` A ) e. RR+ ) -> ( 2 / ( sqrt ` A ) ) e. RR+ ) |
163 |
159 161 162
|
sylancr |
|- ( A e. NN -> ( 2 / ( sqrt ` A ) ) e. RR+ ) |
164 |
163
|
rpge0d |
|- ( A e. NN -> 0 <_ ( 2 / ( sqrt ` A ) ) ) |
165 |
163
|
rpred |
|- ( A e. NN -> ( 2 / ( sqrt ` A ) ) e. RR ) |
166 |
|
subge02 |
|- ( ( 2 e. RR /\ ( 2 / ( sqrt ` A ) ) e. RR ) -> ( 0 <_ ( 2 / ( sqrt ` A ) ) <-> ( 2 - ( 2 / ( sqrt ` A ) ) ) <_ 2 ) ) |
167 |
14 165 166
|
sylancr |
|- ( A e. NN -> ( 0 <_ ( 2 / ( sqrt ` A ) ) <-> ( 2 - ( 2 / ( sqrt ` A ) ) ) <_ 2 ) ) |
168 |
164 167
|
mpbid |
|- ( A e. NN -> ( 2 - ( 2 / ( sqrt ` A ) ) ) <_ 2 ) |
169 |
158 168
|
eqbrtrd |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( 2 / ( sqrt ` ( n - 1 ) ) ) - ( 2 / ( sqrt ` n ) ) ) <_ 2 ) |
170 |
13 23 24 113 169
|
letrd |
|- ( A e. NN -> sum_ n e. ( 2 ... A ) ( ( log ` n ) / ( n x. ( n - 1 ) ) ) <_ 2 ) |