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Theorem selberg2lem 22684
Description: Lemma for selberg2 22685. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg2lem  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O(1)
Distinct variable group:    x, n

Proof of Theorem selberg2lem
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 rpre 10985 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
2 chpcl 22347 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
31, 2syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
43recnd 9400 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
5 rprege0 10993 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
6 flge0nn0 11650 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e. 
NN0 )
8 nn0p1nn 10607 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  NN0  ->  ( ( |_ `  x )  +  1 )  e.  NN )
97, 8syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  NN )
109nnrpd 11014 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  RR+ )
1110relogcld 21957 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  RR )
1211recnd 9400 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  CC )
13 relogcl 21912 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1413recnd 9400 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1512, 14subcld 9707 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
164, 15mulcld 9394 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  CC )
17 fzfid 11779 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
18 elfznn 11465 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1918adantl 463 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
2019nnrpd 11014 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
21 1rp 10983 . . . . . . . . . . . . 13  |-  1  e.  RR+
22 rpaddcl 10999 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR+  /\  1  e.  RR+ )  ->  (
n  +  1 )  e.  RR+ )
2321, 22mpan2 664 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  +  1 )  e.  RR+ )
2423relogcld 21957 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  ( n  +  1 ) )  e.  RR )
25 relogcl 21912 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2624, 25resubcld 9764 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  e.  RR )
27 rpre 10985 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
28 chpcl 22347 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  (ψ `  n )  e.  RR )
2927, 28syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  RR )
3026, 29remulcld 9402 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  RR )
3130recnd 9400 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
3220, 31syl 16 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
3317, 32fsumcl 13194 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC )
34 rpcnne0 10996 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
35 divsubdir 10015 . . . . . 6  |-  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  e.  CC  /\  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 ) )  ->  ( (
( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  /  x
)  =  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  /  x )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) ) )
3616, 33, 34, 35syl3anc 1211 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  /  x
)  =  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  /  x )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) ) )
374, 12mulcld 9394 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  e.  CC )
384, 14mulcld 9394 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
3937, 38, 33sub32d 9739 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  =  ( ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )
404, 12, 14subdid 9788 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) ) )
4140oveq1d 6095 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) ) )
42 fveq2 5679 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( log `  m )  =  ( log `  n
) )
43 oveq1 6087 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  -  1 )  =  ( n  - 
1 ) )
4443fveq2d 5683 . . . . . . . . . . 11  |-  ( m  =  n  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) )
4542, 44jca 529 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( log `  m
)  =  ( log `  n )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) ) )
46 fveq2 5679 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  ( log `  m )  =  ( log `  (
n  +  1 ) ) )
47 oveq1 6087 . . . . . . . . . . . 12  |-  ( m  =  ( n  + 
1 )  ->  (
m  -  1 )  =  ( ( n  +  1 )  - 
1 ) )
4847fveq2d 5683 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) )
4946, 48jca 529 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
( log `  m
)  =  ( log `  ( n  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) )
50 fveq2 5679 . . . . . . . . . . . 12  |-  ( m  =  1  ->  ( log `  m )  =  ( log `  1
) )
51 log1 21919 . . . . . . . . . . . 12  |-  ( log `  1 )  =  0
5250, 51syl6eq 2481 . . . . . . . . . . 11  |-  ( m  =  1  ->  ( log `  m )  =  0 )
53 oveq1 6087 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
m  -  1 )  =  ( 1  -  1 ) )
54 1m1e0 10378 . . . . . . . . . . . . . 14  |-  ( 1  -  1 )  =  0
5553, 54syl6eq 2481 . . . . . . . . . . . . 13  |-  ( m  =  1  ->  (
m  -  1 )  =  0 )
5655fveq2d 5683 . . . . . . . . . . . 12  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  0 ) )
57 2pos 10401 . . . . . . . . . . . . 13  |-  0  <  2
58 0re 9374 . . . . . . . . . . . . . 14  |-  0  e.  RR
59 chpeq0 22432 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  (
(ψ `  0 )  =  0  <->  0  <  2 ) )
6058, 59ax-mp 5 . . . . . . . . . . . . 13  |-  ( (ψ `  0 )  =  0  <->  0  <  2
)
6157, 60mpbir 209 . . . . . . . . . . . 12  |-  (ψ ` 
0 )  =  0
6256, 61syl6eq 2481 . . . . . . . . . . 11  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  0 )
6352, 62jca 529 . . . . . . . . . 10  |-  ( m  =  1  ->  (
( log `  m
)  =  0  /\  (ψ `  ( m  -  1 ) )  =  0 ) )
64 fveq2 5679 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  ( log `  m )  =  ( log `  (
( |_ `  x
)  +  1 ) ) )
65 oveq1 6087 . . . . . . . . . . . 12  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
m  -  1 )  =  ( ( ( |_ `  x )  +  1 )  - 
1 ) )
6665fveq2d 5683 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )
6764, 66jca 529 . . . . . . . . . 10  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
( log `  m
)  =  ( log `  ( ( |_ `  x )  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) ) )
68 nnuz 10884 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
699, 68syl6eleq 2523 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  ( ZZ>= `  1 )
)
70 elfznn 11465 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... ( ( |_ `  x )  +  1 ) )  ->  m  e.  NN )
7170adantl 463 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  NN )
7271nnrpd 11014 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR+ )
7372relogcld 21957 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  RR )
7473recnd 9400 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  CC )
7571nnred 10325 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR )
76 peano2rem 9663 . . . . . . . . . . . . 13  |-  ( m  e.  RR  ->  (
m  -  1 )  e.  RR )
7775, 76syl 16 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (
m  -  1 )  e.  RR )
78 chpcl 22347 . . . . . . . . . . . 12  |-  ( ( m  -  1 )  e.  RR  ->  (ψ `  ( m  -  1 ) )  e.  RR )
7977, 78syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  RR )
8079recnd 9400 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  CC )
8145, 49, 63, 67, 69, 74, 80fsumparts 13252 . . . . . . . . 9  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1..^ ( ( |_ `  x )  +  1 ) ) ( ( log `  n
)  x.  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) ) )  =  ( ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  -  sum_ n  e.  ( 1..^ ( ( |_ `  x
)  +  1 ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  x.  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) ) )
827nn0zd 10733 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  ZZ )
83 fzval3 11589 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  ZZ  ->  (
1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8482, 83syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8584eqcomd 2438 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1..^ ( ( |_ `  x )  +  1 ) )  =  ( 1 ... ( |_
`  x ) ) )
8619nncnd 10326 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  CC )
87 ax-1cn 9328 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
88 pncan 9604 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  + 
1 )  -  1 )  =  n )
8986, 87, 88sylancl 655 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  n )
90 npcan 9607 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
9186, 87, 90sylancl 655 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  -  1 )  +  1 )  =  n )
9289, 91eqtr4d 2468 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  ( ( n  - 
1 )  +  1 ) )
9392fveq2d 5683 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  ( ( n  - 
1 )  +  1 ) ) )
94 nnm1nn0 10609 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
9519, 94syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e. 
NN0 )
96 chpp1 22378 . . . . . . . . . . . . . . . 16  |-  ( ( n  -  1 )  e.  NN0  ->  (ψ `  ( ( n  - 
1 )  +  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  (
( n  -  1 )  +  1 ) ) ) )
9795, 96syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  -  1 )  +  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) ) )
9891fveq2d 5683 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  (
( n  -  1 )  +  1 ) )  =  (Λ `  n
) )
9998oveq2d 6096 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) )  =  ( (ψ `  (
n  -  1 ) )  +  (Λ `  n
) ) )
10093, 97, 993eqtrd 2469 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  n ) ) )
101100oveq1d 6095 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  ( ( (ψ `  ( n  -  1
) )  +  (Λ `  n ) )  -  (ψ `  ( n  - 
1 ) ) ) )
10295nn0red 10625 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e.  RR )
103 chpcl 22347 . . . . . . . . . . . . . . . 16  |-  ( ( n  -  1 )  e.  RR  ->  (ψ `  ( n  -  1 ) )  e.  RR )
104102, 103syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  RR )
105104recnd 9400 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  CC )
106 vmacl 22341 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
10719, 106syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
108107recnd 9400 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
109105, 108pncan2d 9709 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(ψ `  ( n  -  1 ) )  +  (Λ `  n
) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
110101, 109eqtrd 2465 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
111110oveq2d 6096 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
11220relogcld 21957 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
113112recnd 9400 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
114108, 113mulcomd 9395 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
115111, 114eqtr4d 2468 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( (Λ `  n
)  x.  ( log `  n ) ) )
11685, 115sumeq12rdv 13168 . . . . . . . . 9  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1..^ ( ( |_ `  x )  +  1 ) ) ( ( log `  n
)  x.  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) ) )
1177nn0cnd 10626 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  CC )
118 pncan 9604 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( |_
`  x )  +  1 )  -  1 )  =  ( |_
`  x ) )
119117, 87, 118sylancl 655 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  -  1 )  =  ( |_ `  x
) )
120119fveq2d 5683 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  ( |_ `  x
) ) )
121 chpfl 22373 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR  ->  (ψ `  ( |_ `  x
) )  =  (ψ `  x ) )
1221, 121syl 16 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( |_ `  x ) )  =  (ψ `  x ) )
123120, 122eqtrd 2465 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  x ) )
124123oveq2d 6096 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  x ) ) )
12512, 4mulcomd 9395 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  x
) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
126124, 125eqtrd 2465 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
127 0cn 9366 . . . . . . . . . . . . . 14  |-  0  e.  CC
128127mul01i 9547 . . . . . . . . . . . . 13  |-  ( 0  x.  0 )  =  0
129128a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 0  x.  0 )  =  0 )
130126, 129oveq12d 6098 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  - 
0 ) )
13137subid1d 9696 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) )  -  0 )  =  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
132130, 131eqtrd 2465 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
13389fveq2d 5683 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  n ) )
134133oveq2d 6096 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  ( (
n  +  1 )  -  1 ) ) )  =  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )
13585, 134sumeq12rdv 13168 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1..^ ( ( |_ `  x )  +  1 ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  (
( n  +  1 )  -  1 ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )
136132, 135oveq12d 6098 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  -  sum_ n  e.  ( 1..^ ( ( |_ `  x
)  +  1 ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  x.  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) ) )
13781, 116, 1363eqtr3d 2473 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) ) )
138137oveq1d 6095 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  =  ( ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) ) )
13939, 41, 1383eqtr4d 2475 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )
140139oveq1d 6095 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  /  x
)  =  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )
141 div23 10001 . . . . . . 7  |-  ( ( (ψ `  x )  e.  CC  /\  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 ) )  ->  ( (
(ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
1424, 15, 34, 141syl3anc 1211 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
143142oveq1d 6095 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  /  x )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) )  =  ( ( ( (ψ `  x
)  /  x )  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) ) )
14436, 140, 1433eqtr3rd 2474 . . . 4  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  /  x )  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) )  =  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )
145144mpteq2ia 4362 . . 3  |-  ( x  e.  RR+  |->  ( ( ( (ψ `  x
)  /  x )  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x ) ) )  =  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )
146 ovex 6105 . . . . 5  |-  ( ( (ψ `  x )  /  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  _V
147146a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( (ψ `  x )  /  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  _V )
148 ovex 6105 . . . . 5  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x )  e.  _V
149148a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  /  x )  e.  _V )
150 reex 9361 . . . . . . . 8  |-  RR  e.  _V
151 rpssre 10989 . . . . . . . 8  |-  RR+  C_  RR
152150, 151ssexi 4425 . . . . . . 7  |-  RR+  e.  _V
153152a1i 11 . . . . . 6  |-  ( T. 
->  RR+  e.  _V )
154 ovex 6105 . . . . . . 7  |-  ( (ψ `  x )  /  x
)  e.  _V
155154a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  _V )
15615adantl 463 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
157 eqidd 2434 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) ) )
158 eqidd 2434 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )
159153, 155, 156, 157, 158offval2 6325 . . . . 5  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  oF  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) ) ) )
160 chpo1ub 22614 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
161 0red 9375 . . . . . . . 8  |-  ( T. 
->  0  e.  RR )
162 1red 9389 . . . . . . . 8  |-  ( T. 
->  1  e.  RR )
163 divrcnv 13298 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
16487, 163mp1i 12 . . . . . . . 8  |-  ( T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
165 rpreccl 11002 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
166165rpred 11015 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR )
167166adantl 463 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
16811, 13resubcld 9764 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
169168adantl 463 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
170 rpaddcl 10999 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
17121, 170mpan2 664 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  +  1 )  e.  RR+ )
172171relogcld 21957 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( x  +  1 ) )  e.  RR )
173172, 13resubcld 9764 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  e.  RR )
1747nn0red 10625 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  RR )
175 1red 9389 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  1  e.  RR )
176 flle 11633 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( |_ `  x )  <_  x )
1771, 176syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  <_  x )
178174, 1, 175, 177leadd1dd 9941 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  <_ 
( x  +  1 ) )
17910, 171logled 21961 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  <_  ( x  + 
1 )  <->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) ) )
180178, 179mpbid 210 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) )
18111, 172, 13, 180lesub1dd 9943 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
( log `  (
x  +  1 ) )  -  ( log `  x ) ) )
182 logdifbnd 22272 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
183168, 173, 166, 181, 182letrd 9516 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
184183ad2antrl 720 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) )  <_ 
( 1  /  x
) )
185 fllep1 11635 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <_  ( ( |_ `  x )  +  1 ) )
1861, 185syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  <_ 
( ( |_ `  x )  +  1 ) )
187 logleb 21937 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  (
( |_ `  x
)  +  1 )  e.  RR+ )  ->  (
x  <_  ( ( |_ `  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
18810, 187mpdan 661 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  <_  ( ( |_
`  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
189186, 188mpbid 210 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) )
19011, 13subge0d 9917 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 0  <_  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) )  <-> 
( log `  x
)  <_  ( log `  ( ( |_ `  x )  +  1 ) ) ) )
191189, 190mpbird 232 . . . . . . . . 9  |-  ( x  e.  RR+  ->  0  <_ 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )
192191ad2antrl 720 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( ( log `  ( ( |_
`  x )  +  1 ) )  -  ( log `  x ) ) )
193161, 162, 164, 167, 169, 184, 192rlimsqz2 13112 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  ~~> r  0 )
194 rlimo1 13078 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  ~~> r  0  ->  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  e.  O(1) )
195193, 194syl 16 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  O(1) )
196 o1mul 13076 . . . . . 6  |-  ( ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  e.  O(1)  /\  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  e.  O(1) )  -> 
( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  oF  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O(1) )
197160, 195, 196sylancr 656 . . . . 5  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  oF  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O(1) )
198159, 197eqeltrrd 2508 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )  e.  O(1) )
199 nnrp 10988 . . . . . . . . 9  |-  ( m  e.  NN  ->  m  e.  RR+ )
200199ssriv 3348 . . . . . . . 8  |-  NN  C_  RR+
201200a1i 11 . . . . . . 7  |-  ( T. 
->  NN  C_  RR+ )
202201sselda 3344 . . . . . 6  |-  ( ( T.  /\  n  e.  NN )  ->  n  e.  RR+ )
203202, 31syl 16 . . . . 5  |-  ( ( T.  /\  n  e.  NN )  ->  (
( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
204 chpo1ub 22614 . . . . . . . 8  |-  ( n  e.  RR+  |->  ( (ψ `  n )  /  n
) )  e.  O(1)
205204a1i 11 . . . . . . 7  |-  ( T. 
->  ( n  e.  RR+  |->  ( (ψ `  n )  /  n ) )  e.  O(1) )
206 rerpdivcl 11006 . . . . . . . . 9  |-  ( ( (ψ `  n )  e.  RR  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20729, 206mpancom 662 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  e.  RR )
208207adantl 463 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20931adantl 463 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
210 rpreccl 11002 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR+ )
211210rpred 11015 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR )
212 chpge0 22349 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  0  <_  (ψ `  n )
)
21327, 212syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
(ψ `  n )
)
214 logdifbnd 22272 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  <_  (
1  /  n ) )
21526, 211, 29, 213, 214lemul1ad 10260 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  <_  ( (
1  /  n )  x.  (ψ `  n
) ) )
21627lep1d 10252 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  n  <_ 
( n  +  1 ) )
217 logleb 21937 . . . . . . . . . . . . . 14  |-  ( ( n  e.  RR+  /\  (
n  +  1 )  e.  RR+ )  ->  (
n  <_  ( n  +  1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
21823, 217mpdan 661 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  ( n  <_  ( n  + 
1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
219216, 218mpbid 210 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) )
22024, 25subge0d 9917 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( 0  <_  ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  <-> 
( log `  n
)  <_  ( log `  ( n  +  1 ) ) ) )
221219, 220mpbird 232 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( ( log `  (
n  +  1 ) )  -  ( log `  n ) ) )
22226, 29, 221, 213mulge0d 9904 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
22330, 222absidd 12893 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
224 rpregt0 10992 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  e.  RR  /\  0  <  n ) )
225 divge0 10186 . . . . . . . . . . . 12  |-  ( ( ( (ψ `  n
)  e.  RR  /\  0  <_  (ψ `  n
) )  /\  (
n  e.  RR  /\  0  <  n ) )  ->  0  <_  (
(ψ `  n )  /  n ) )
22629, 213, 224, 225syl21anc 1210 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( (ψ `  n
)  /  n ) )
227207, 226absidd 12893 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( (ψ `  n )  /  n
) )
22829recnd 9400 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  CC )
229 rpcn 10987 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  CC )
230 rpne0 10994 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  =/=  0 )
231228, 229, 230divrec2d 10099 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
232227, 231eqtrd 2465 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
233215, 223, 2323brtr4d 4310 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  <_ 
( abs `  (
(ψ `  n )  /  n ) ) )
234233ad2antrl 720 . . . . . . 7  |-  ( ( T.  /\  ( n  e.  RR+  /\  1  <_  n ) )  -> 
( abs `  (
( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  <_  ( abs `  ( (ψ `  n )  /  n
) ) )
235162, 205, 208, 209, 234o1le 13114 . . . . . 6  |-  ( T. 
->  ( n  e.  RR+  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
236201, 235o1res2 13025 . . . . 5  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
237203, 236o1fsum 13259 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  /  x
) )  e.  O(1) )
238147, 149, 198, 237o1sub2 13087 . . 3  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  -  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  /  x
) ) )  e.  O(1) )
239145, 238syl5eqelr 2518 . 2  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O(1) )
240239trud 1371 1  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1362   T. wtru 1363    e. wcel 1755    =/= wne 2596   _Vcvv 2962    C_ wss 3316   class class class wbr 4280    e. cmpt 4338   ` cfv 5406  (class class class)co 6080    oFcof 6307   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   NN0cn0 10567   ZZcz 10634   ZZ>=cuz 10849   RR+crp 10979   ...cfz 11424  ..^cfzo 11532   |_cfl 11624   abscabs 12707    ~~> r crli 12947   O(1)co1 12948   sum_csu 13147   logclog 21891  Λcvma 22314  ψcchp 22315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-o1 12952  df-lo1 12953  df-sum 13148  df-ef 13336  df-e 13337  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894  df-cht 22319  df-vma 22320  df-chp 22321  df-ppi 22322
This theorem is referenced by:  selberg2  22685  selberg3lem2  22692
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