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Theorem selberg2lem 22931
Description: Lemma for selberg2 22932. 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 11107 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
2 chpcl 22594 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
31, 2syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
43recnd 9522 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
5 rprege0 11115 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
6 flge0nn0 11782 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e. 
NN0 )
8 nn0p1nn 10729 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  NN0  ->  ( ( |_ `  x )  +  1 )  e.  NN )
97, 8syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  NN )
109nnrpd 11136 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  RR+ )
1110relogcld 22204 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  RR )
1211recnd 9522 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  CC )
13 relogcl 22159 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1413recnd 9522 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1512, 14subcld 9829 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
164, 15mulcld 9516 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  CC )
17 fzfid 11911 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
18 elfznn 11594 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1918adantl 466 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
2019nnrpd 11136 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
21 1rp 11105 . . . . . . . . . . . . 13  |-  1  e.  RR+
22 rpaddcl 11121 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR+  /\  1  e.  RR+ )  ->  (
n  +  1 )  e.  RR+ )
2321, 22mpan2 671 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  +  1 )  e.  RR+ )
2423relogcld 22204 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  ( n  +  1 ) )  e.  RR )
25 relogcl 22159 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2624, 25resubcld 9886 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  e.  RR )
27 rpre 11107 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
28 chpcl 22594 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  (ψ `  n )  e.  RR )
2927, 28syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  RR )
3026, 29remulcld 9524 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  RR )
3130recnd 9522 . . . . . . . 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 13327 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC )
34 rpcnne0 11118 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
35 divsubdir 10137 . . . . . 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 1219 . . . . 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 9516 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  e.  CC )
384, 14mulcld 9516 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
3937, 38, 33sub32d 9861 . . . . . . 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 9910 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) ) )
4140oveq1d 6214 . . . . . . 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 5798 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( log `  m )  =  ( log `  n
) )
43 oveq1 6206 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  -  1 )  =  ( n  - 
1 ) )
4443fveq2d 5802 . . . . . . . . . . 11  |-  ( m  =  n  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) )
4542, 44jca 532 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( log `  m
)  =  ( log `  n )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) ) )
46 fveq2 5798 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  ( log `  m )  =  ( log `  (
n  +  1 ) ) )
47 oveq1 6206 . . . . . . . . . . . 12  |-  ( m  =  ( n  + 
1 )  ->  (
m  -  1 )  =  ( ( n  +  1 )  - 
1 ) )
4847fveq2d 5802 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) )
4946, 48jca 532 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
( log `  m
)  =  ( log `  ( n  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) )
50 fveq2 5798 . . . . . . . . . . . 12  |-  ( m  =  1  ->  ( log `  m )  =  ( log `  1
) )
51 log1 22166 . . . . . . . . . . . 12  |-  ( log `  1 )  =  0
5250, 51syl6eq 2511 . . . . . . . . . . 11  |-  ( m  =  1  ->  ( log `  m )  =  0 )
53 oveq1 6206 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
m  -  1 )  =  ( 1  -  1 ) )
54 1m1e0 10500 . . . . . . . . . . . . . 14  |-  ( 1  -  1 )  =  0
5553, 54syl6eq 2511 . . . . . . . . . . . . 13  |-  ( m  =  1  ->  (
m  -  1 )  =  0 )
5655fveq2d 5802 . . . . . . . . . . . 12  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  0 ) )
57 2pos 10523 . . . . . . . . . . . . 13  |-  0  <  2
58 0re 9496 . . . . . . . . . . . . . 14  |-  0  e.  RR
59 chpeq0 22679 . . . . . . . . . . . . . 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 2511 . . . . . . . . . . 11  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  0 )
6352, 62jca 532 . . . . . . . . . 10  |-  ( m  =  1  ->  (
( log `  m
)  =  0  /\  (ψ `  ( m  -  1 ) )  =  0 ) )
64 fveq2 5798 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  ( log `  m )  =  ( log `  (
( |_ `  x
)  +  1 ) ) )
65 oveq1 6206 . . . . . . . . . . . 12  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
m  -  1 )  =  ( ( ( |_ `  x )  +  1 )  - 
1 ) )
6665fveq2d 5802 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )
6764, 66jca 532 . . . . . . . . . 10  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
( log `  m
)  =  ( log `  ( ( |_ `  x )  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) ) )
68 nnuz 11006 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
699, 68syl6eleq 2552 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  ( ZZ>= `  1 )
)
70 elfznn 11594 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... ( ( |_ `  x )  +  1 ) )  ->  m  e.  NN )
7170adantl 466 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  NN )
7271nnrpd 11136 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR+ )
7372relogcld 22204 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  RR )
7473recnd 9522 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  CC )
7571nnred 10447 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR )
76 peano2rem 9785 . . . . . . . . . . . . 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 22594 . . . . . . . . . . . 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 9522 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  CC )
8145, 49, 63, 67, 69, 74, 80fsumparts 13386 . . . . . . . . 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 10855 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  ZZ )
83 fzval3 11721 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  ZZ  ->  (
1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8482, 83syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8584eqcomd 2462 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1..^ ( ( |_ `  x )  +  1 ) )  =  ( 1 ... ( |_
`  x ) ) )
8619nncnd 10448 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  CC )
87 ax-1cn 9450 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
88 pncan 9726 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  + 
1 )  -  1 )  =  n )
8986, 87, 88sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  n )
90 npcan 9729 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
9186, 87, 90sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  -  1 )  +  1 )  =  n )
9289, 91eqtr4d 2498 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  ( ( n  - 
1 )  +  1 ) )
9392fveq2d 5802 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  ( ( n  - 
1 )  +  1 ) ) )
94 nnm1nn0 10731 . . . . . . . . . . . . . . . . 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 22625 . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  (
( n  -  1 )  +  1 ) )  =  (Λ `  n
) )
9998oveq2d 6215 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) )  =  ( (ψ `  (
n  -  1 ) )  +  (Λ `  n
) ) )
10093, 97, 993eqtrd 2499 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  n ) ) )
101100oveq1d 6214 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  ( ( (ψ `  ( n  -  1
) )  +  (Λ `  n ) )  -  (ψ `  ( n  - 
1 ) ) ) )
10295nn0red 10747 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e.  RR )
103 chpcl 22594 . . . . . . . . . . . . . . . 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 9522 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  CC )
106 vmacl 22588 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
10719, 106syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
108107recnd 9522 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
109105, 108pncan2d 9831 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(ψ `  ( n  -  1 ) )  +  (Λ `  n
) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
110101, 109eqtrd 2495 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
111110oveq2d 6215 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
11220relogcld 22204 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
113112recnd 9522 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
114108, 113mulcomd 9517 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
115111, 114eqtr4d 2498 . . . . . . . . . 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 13301 . . . . . . . . 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 10748 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  CC )
118 pncan 9726 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( |_
`  x )  +  1 )  -  1 )  =  ( |_
`  x ) )
119117, 87, 118sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  -  1 )  =  ( |_ `  x
) )
120119fveq2d 5802 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  ( |_ `  x
) ) )
121 chpfl 22620 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR  ->  (ψ `  ( |_ `  x
) )  =  (ψ `  x ) )
1221, 121syl 16 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( |_ `  x ) )  =  (ψ `  x ) )
123120, 122eqtrd 2495 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  x ) )
124123oveq2d 6215 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  x ) ) )
12512, 4mulcomd 9517 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  x
) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
126124, 125eqtrd 2495 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
127 0cn 9488 . . . . . . . . . . . . . 14  |-  0  e.  CC
128127mul01i 9669 . . . . . . . . . . . . 13  |-  ( 0  x.  0 )  =  0
129128a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 0  x.  0 )  =  0 )
130126, 129oveq12d 6217 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  - 
0 ) )
13137subid1d 9818 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) )  -  0 )  =  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
132130, 131eqtrd 2495 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
13389fveq2d 5802 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  n ) )
134133oveq2d 6215 . . . . . . . . . . 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 13301 . . . . . . . . . 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 6217 . . . . . . . . 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 2503 . . . . . . . 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 6214 . . . . . . 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 2505 . . . . . 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 6214 . . . . 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 10123 . . . . . . 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 1219 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
143142oveq1d 6214 . . . . 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 2504 . . . 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 4481 . . 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 6224 . . . . 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 6224 . . . . 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 9483 . . . . . . . 8  |-  RR  e.  _V
151 rpssre 11111 . . . . . . . 8  |-  RR+  C_  RR
152150, 151ssexi 4544 . . . . . . 7  |-  RR+  e.  _V
153152a1i 11 . . . . . 6  |-  ( T. 
->  RR+  e.  _V )
154 ovex 6224 . . . . . . 7  |-  ( (ψ `  x )  /  x
)  e.  _V
155154a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  _V )
15615adantl 466 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
157 eqidd 2455 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) ) )
158 eqidd 2455 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )
159153, 155, 156, 157, 158offval2 6445 . . . . 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 22861 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
161 0red 9497 . . . . . . . 8  |-  ( T. 
->  0  e.  RR )
162 1red 9511 . . . . . . . 8  |-  ( T. 
->  1  e.  RR )
163 divrcnv 13432 . . . . . . . . 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 11124 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
166165rpred 11137 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR )
167166adantl 466 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
16811, 13resubcld 9886 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
169168adantl 466 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
170 rpaddcl 11121 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
17121, 170mpan2 671 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  +  1 )  e.  RR+ )
172171relogcld 22204 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( x  +  1 ) )  e.  RR )
173172, 13resubcld 9886 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  e.  RR )
1747nn0red 10747 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  RR )
175 1red 9511 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  1  e.  RR )
176 flle 11765 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( |_ `  x )  <_  x )
1771, 176syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  <_  x )
178174, 1, 175, 177leadd1dd 10063 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  <_ 
( x  +  1 ) )
17910, 171logled 22208 . . . . . . . . . . . 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 10065 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
( log `  (
x  +  1 ) )  -  ( log `  x ) ) )
182 logdifbnd 22519 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
183168, 173, 166, 181, 182letrd 9638 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
184183ad2antrl 727 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) )  <_ 
( 1  /  x
) )
185 fllep1 11767 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <_  ( ( |_ `  x )  +  1 ) )
1861, 185syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  <_ 
( ( |_ `  x )  +  1 ) )
187 logleb 22184 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  (
( |_ `  x
)  +  1 )  e.  RR+ )  ->  (
x  <_  ( ( |_ `  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
18810, 187mpdan 668 . . . . . . . . . . 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 10039 . . . . . . . . . 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 727 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( ( log `  ( ( |_
`  x )  +  1 ) )  -  ( log `  x ) ) )
193161, 162, 164, 167, 169, 184, 192rlimsqz2 13245 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  ~~> r  0 )
194 rlimo1 13211 . . . . . . 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 13209 . . . . . 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 663 . . . . 5  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  oF  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O(1) )
198159, 197eqeltrrd 2543 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )  e.  O(1) )
199 nnrp 11110 . . . . . . . . 9  |-  ( m  e.  NN  ->  m  e.  RR+ )
200199ssriv 3467 . . . . . . . 8  |-  NN  C_  RR+
201200a1i 11 . . . . . . 7  |-  ( T. 
->  NN  C_  RR+ )
202201sselda 3463 . . . . . 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 22861 . . . . . . . 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 11128 . . . . . . . . 9  |-  ( ( (ψ `  n )  e.  RR  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20729, 206mpancom 669 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  e.  RR )
208207adantl 466 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20931adantl 466 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
210 rpreccl 11124 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR+ )
211210rpred 11137 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR )
212 chpge0 22596 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  0  <_  (ψ `  n )
)
21327, 212syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
(ψ `  n )
)
214 logdifbnd 22519 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  <_  (
1  /  n ) )
21526, 211, 29, 213, 214lemul1ad 10382 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  <_  ( (
1  /  n )  x.  (ψ `  n
) ) )
21627lep1d 10374 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  n  <_ 
( n  +  1 ) )
217 logleb 22184 . . . . . . . . . . . . . 14  |-  ( ( n  e.  RR+  /\  (
n  +  1 )  e.  RR+ )  ->  (
n  <_  ( n  +  1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
21823, 217mpdan 668 . . . . . . . . . . . . 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 10039 . . . . . . . . . . . 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 10026 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
22330, 222absidd 13026 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
224 rpregt0 11114 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  e.  RR  /\  0  <  n ) )
225 divge0 10308 . . . . . . . . . . . 12  |-  ( ( ( (ψ `  n
)  e.  RR  /\  0  <_  (ψ `  n
) )  /\  (
n  e.  RR  /\  0  <  n ) )  ->  0  <_  (
(ψ `  n )  /  n ) )
22629, 213, 224, 225syl21anc 1218 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( (ψ `  n
)  /  n ) )
227207, 226absidd 13026 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( (ψ `  n )  /  n
) )
22829recnd 9522 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  CC )
229 rpcn 11109 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  CC )
230 rpne0 11116 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  =/=  0 )
231228, 229, 230divrec2d 10221 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
232227, 231eqtrd 2495 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
233215, 223, 2323brtr4d 4429 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  <_ 
( abs `  (
(ψ `  n )  /  n ) ) )
234233ad2antrl 727 . . . . . . 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 13247 . . . . . 6  |-  ( T. 
->  ( n  e.  RR+  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
236201, 235o1res2 13158 . . . . 5  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
237203, 236o1fsum 13393 . . . 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 13220 . . 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 2547 . 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 1379 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 1370   T. wtru 1371    e. wcel 1758    =/= wne 2647   _Vcvv 3076    C_ wss 3435   class class class wbr 4399    |-> cmpt 4457   ` cfv 5525  (class class class)co 6199    oFcof 6427   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397    < clt 9528    <_ cle 9529    - cmin 9705    / cdiv 10103   NNcn 10432   2c2 10481   NN0cn0 10689   ZZcz 10756   ZZ>=cuz 10971   RR+crp 11101   ...cfz 11553  ..^cfzo 11664   |_cfl 11756   abscabs 12840    ~~> r crli 13080   O(1)co1 13081   sum_csu 13280   logclog 22138  Λcvma 22561  ψcchp 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ioc 11415  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-fac 12168  df-bc 12195  df-hash 12220  df-shft 12673  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-limsup 13066  df-clim 13083  df-rlim 13084  df-o1 13085  df-lo1 13086  df-sum 13281  df-ef 13470  df-e 13471  df-sin 13472  df-cos 13473  df-pi 13475  df-dvds 13653  df-gcd 13808  df-prm 13881  df-pc 14021  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-lp 18871  df-perf 18872  df-cn 18962  df-cnp 18963  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cncf 20585  df-limc 21473  df-dv 21474  df-log 22140  df-cxp 22141  df-cht 22566  df-vma 22567  df-chp 22568  df-ppi 22569
This theorem is referenced by:  selberg2  22932  selberg3lem2  22939
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