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Theorem selberg2lem 22774
Description: Lemma for selberg2 22775. 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 10989 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
2 chpcl 22437 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
31, 2syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
43recnd 9404 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
5 rprege0 10997 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
6 flge0nn0 11658 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e. 
NN0 )
8 nn0p1nn 10611 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  NN0  ->  ( ( |_ `  x )  +  1 )  e.  NN )
97, 8syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  NN )
109nnrpd 11018 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  RR+ )
1110relogcld 22047 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  RR )
1211recnd 9404 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  CC )
13 relogcl 22002 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1413recnd 9404 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1512, 14subcld 9711 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
164, 15mulcld 9398 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  CC )
17 fzfid 11787 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
18 elfznn 11470 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1918adantl 466 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
2019nnrpd 11018 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
21 1rp 10987 . . . . . . . . . . . . 13  |-  1  e.  RR+
22 rpaddcl 11003 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR+  /\  1  e.  RR+ )  ->  (
n  +  1 )  e.  RR+ )
2321, 22mpan2 671 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  +  1 )  e.  RR+ )
2423relogcld 22047 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  ( n  +  1 ) )  e.  RR )
25 relogcl 22002 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2624, 25resubcld 9768 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  e.  RR )
27 rpre 10989 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
28 chpcl 22437 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  (ψ `  n )  e.  RR )
2927, 28syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  RR )
3026, 29remulcld 9406 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  RR )
3130recnd 9404 . . . . . . . 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 13202 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC )
34 rpcnne0 11000 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
35 divsubdir 10019 . . . . . 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 1218 . . . . 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 9398 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  e.  CC )
384, 14mulcld 9398 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
3937, 38, 33sub32d 9743 . . . . . . 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 9792 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) ) )
4140oveq1d 6101 . . . . . . 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 5686 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( log `  m )  =  ( log `  n
) )
43 oveq1 6093 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  -  1 )  =  ( n  - 
1 ) )
4443fveq2d 5690 . . . . . . . . . . 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 5686 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  ( log `  m )  =  ( log `  (
n  +  1 ) ) )
47 oveq1 6093 . . . . . . . . . . . 12  |-  ( m  =  ( n  + 
1 )  ->  (
m  -  1 )  =  ( ( n  +  1 )  - 
1 ) )
4847fveq2d 5690 . . . . . . . . . . 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 5686 . . . . . . . . . . . 12  |-  ( m  =  1  ->  ( log `  m )  =  ( log `  1
) )
51 log1 22009 . . . . . . . . . . . 12  |-  ( log `  1 )  =  0
5250, 51syl6eq 2486 . . . . . . . . . . 11  |-  ( m  =  1  ->  ( log `  m )  =  0 )
53 oveq1 6093 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
m  -  1 )  =  ( 1  -  1 ) )
54 1m1e0 10382 . . . . . . . . . . . . . 14  |-  ( 1  -  1 )  =  0
5553, 54syl6eq 2486 . . . . . . . . . . . . 13  |-  ( m  =  1  ->  (
m  -  1 )  =  0 )
5655fveq2d 5690 . . . . . . . . . . . 12  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  0 ) )
57 2pos 10405 . . . . . . . . . . . . 13  |-  0  <  2
58 0re 9378 . . . . . . . . . . . . . 14  |-  0  e.  RR
59 chpeq0 22522 . . . . . . . . . . . . . 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 2486 . . . . . . . . . . 11  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  0 )
6352, 62jca 532 . . . . . . . . . 10  |-  ( m  =  1  ->  (
( log `  m
)  =  0  /\  (ψ `  ( m  -  1 ) )  =  0 ) )
64 fveq2 5686 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  ( log `  m )  =  ( log `  (
( |_ `  x
)  +  1 ) ) )
65 oveq1 6093 . . . . . . . . . . . 12  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
m  -  1 )  =  ( ( ( |_ `  x )  +  1 )  - 
1 ) )
6665fveq2d 5690 . . . . . . . . . . 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 10888 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
699, 68syl6eleq 2528 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  ( ZZ>= `  1 )
)
70 elfznn 11470 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... ( ( |_ `  x )  +  1 ) )  ->  m  e.  NN )
7170adantl 466 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  NN )
7271nnrpd 11018 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR+ )
7372relogcld 22047 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  RR )
7473recnd 9404 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  CC )
7571nnred 10329 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR )
76 peano2rem 9667 . . . . . . . . . . . . 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 22437 . . . . . . . . . . . 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 9404 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  CC )
8145, 49, 63, 67, 69, 74, 80fsumparts 13261 . . . . . . . . 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 10737 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  ZZ )
83 fzval3 11597 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  ZZ  ->  (
1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8482, 83syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8584eqcomd 2443 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1..^ ( ( |_ `  x )  +  1 ) )  =  ( 1 ... ( |_
`  x ) ) )
8619nncnd 10330 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  CC )
87 ax-1cn 9332 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
88 pncan 9608 . . . . . . . . . . . . . . . . . 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 9611 . . . . . . . . . . . . . . . . . 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 2473 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  ( ( n  - 
1 )  +  1 ) )
9392fveq2d 5690 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  ( ( n  - 
1 )  +  1 ) ) )
94 nnm1nn0 10613 . . . . . . . . . . . . . . . . 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 22468 . . . . . . . . . . . . . . . 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 5690 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  (
( n  -  1 )  +  1 ) )  =  (Λ `  n
) )
9998oveq2d 6102 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) )  =  ( (ψ `  (
n  -  1 ) )  +  (Λ `  n
) ) )
10093, 97, 993eqtrd 2474 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  n ) ) )
101100oveq1d 6101 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  ( ( (ψ `  ( n  -  1
) )  +  (Λ `  n ) )  -  (ψ `  ( n  - 
1 ) ) ) )
10295nn0red 10629 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e.  RR )
103 chpcl 22437 . . . . . . . . . . . . . . . 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 9404 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  CC )
106 vmacl 22431 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
10719, 106syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
108107recnd 9404 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
109105, 108pncan2d 9713 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(ψ `  ( n  -  1 ) )  +  (Λ `  n
) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
110101, 109eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
111110oveq2d 6102 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
11220relogcld 22047 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
113112recnd 9404 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
114108, 113mulcomd 9399 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
115111, 114eqtr4d 2473 . . . . . . . . . 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 13176 . . . . . . . . 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 10630 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  CC )
118 pncan 9608 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( |_
`  x )  +  1 )  -  1 )  =  ( |_
`  x ) )
119117, 87, 118sylancl 662 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  -  1 )  =  ( |_ `  x
) )
120119fveq2d 5690 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  ( |_ `  x
) ) )
121 chpfl 22463 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR  ->  (ψ `  ( |_ `  x
) )  =  (ψ `  x ) )
1221, 121syl 16 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( |_ `  x ) )  =  (ψ `  x ) )
123120, 122eqtrd 2470 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  x ) )
124123oveq2d 6102 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  x ) ) )
12512, 4mulcomd 9399 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  x
) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
126124, 125eqtrd 2470 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
127 0cn 9370 . . . . . . . . . . . . . 14  |-  0  e.  CC
128127mul01i 9551 . . . . . . . . . . . . 13  |-  ( 0  x.  0 )  =  0
129128a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 0  x.  0 )  =  0 )
130126, 129oveq12d 6104 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  - 
0 ) )
13137subid1d 9700 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) )  -  0 )  =  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
132130, 131eqtrd 2470 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
13389fveq2d 5690 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  n ) )
134133oveq2d 6102 . . . . . . . . . . 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 13176 . . . . . . . . . 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 6104 . . . . . . . . 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 2478 . . . . . . . 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 6101 . . . . . . 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 2480 . . . . . 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 6101 . . . . 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 10005 . . . . . . 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 1218 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
143142oveq1d 6101 . . . . 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 2479 . . . 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 4369 . . 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 6111 . . . . 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 6111 . . . . 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 9365 . . . . . . . 8  |-  RR  e.  _V
151 rpssre 10993 . . . . . . . 8  |-  RR+  C_  RR
152150, 151ssexi 4432 . . . . . . 7  |-  RR+  e.  _V
153152a1i 11 . . . . . 6  |-  ( T. 
->  RR+  e.  _V )
154 ovex 6111 . . . . . . 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 2439 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) ) )
158 eqidd 2439 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )
159153, 155, 156, 157, 158offval2 6331 . . . . 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 22704 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
161 0red 9379 . . . . . . . 8  |-  ( T. 
->  0  e.  RR )
162 1red 9393 . . . . . . . 8  |-  ( T. 
->  1  e.  RR )
163 divrcnv 13307 . . . . . . . . 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 11006 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
166165rpred 11019 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR )
167166adantl 466 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
16811, 13resubcld 9768 . . . . . . . . 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 11003 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
17121, 170mpan2 671 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  +  1 )  e.  RR+ )
172171relogcld 22047 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( x  +  1 ) )  e.  RR )
173172, 13resubcld 9768 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  e.  RR )
1747nn0red 10629 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  RR )
175 1red 9393 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  1  e.  RR )
176 flle 11641 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( |_ `  x )  <_  x )
1771, 176syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  <_  x )
178174, 1, 175, 177leadd1dd 9945 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  <_ 
( x  +  1 ) )
17910, 171logled 22051 . . . . . . . . . . . 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 9947 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
( log `  (
x  +  1 ) )  -  ( log `  x ) ) )
182 logdifbnd 22362 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
183168, 173, 166, 181, 182letrd 9520 . . . . . . . . 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 11643 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <_  ( ( |_ `  x )  +  1 ) )
1861, 185syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  <_ 
( ( |_ `  x )  +  1 ) )
187 logleb 22027 . . . . . . . . . . . 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 9921 . . . . . . . . . 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 13120 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  ~~> r  0 )
194 rlimo1 13086 . . . . . . 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 13084 . . . . . 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 2513 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )  e.  O(1) )
199 nnrp 10992 . . . . . . . . 9  |-  ( m  e.  NN  ->  m  e.  RR+ )
200199ssriv 3355 . . . . . . . 8  |-  NN  C_  RR+
201200a1i 11 . . . . . . 7  |-  ( T. 
->  NN  C_  RR+ )
202201sselda 3351 . . . . . 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 22704 . . . . . . . 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 11010 . . . . . . . . 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 11006 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR+ )
211210rpred 11019 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR )
212 chpge0 22439 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  0  <_  (ψ `  n )
)
21327, 212syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
(ψ `  n )
)
214 logdifbnd 22362 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  <_  (
1  /  n ) )
21526, 211, 29, 213, 214lemul1ad 10264 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  <_  ( (
1  /  n )  x.  (ψ `  n
) ) )
21627lep1d 10256 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  n  <_ 
( n  +  1 ) )
217 logleb 22027 . . . . . . . . . . . . . 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 9921 . . . . . . . . . . . 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 9908 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
22330, 222absidd 12901 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
224 rpregt0 10996 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  e.  RR  /\  0  <  n ) )
225 divge0 10190 . . . . . . . . . . . 12  |-  ( ( ( (ψ `  n
)  e.  RR  /\  0  <_  (ψ `  n
) )  /\  (
n  e.  RR  /\  0  <  n ) )  ->  0  <_  (
(ψ `  n )  /  n ) )
22629, 213, 224, 225syl21anc 1217 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( (ψ `  n
)  /  n ) )
227207, 226absidd 12901 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( (ψ `  n )  /  n
) )
22829recnd 9404 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  CC )
229 rpcn 10991 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  CC )
230 rpne0 10998 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  =/=  0 )
231228, 229, 230divrec2d 10103 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
232227, 231eqtrd 2470 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
233215, 223, 2323brtr4d 4317 . . . . . . . 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 13122 . . . . . 6  |-  ( T. 
->  ( n  e.  RR+  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
236201, 235o1res2 13033 . . . . 5  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
237203, 236o1fsum 13268 . . . 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 13095 . . 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 2523 . 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 1378 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 1369   T. wtru 1370    e. wcel 1756    =/= wne 2601   _Vcvv 2967    C_ wss 3323   class class class wbr 4287    e. cmpt 4345   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   2c2 10363   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   RR+crp 10983   ...cfz 11429  ..^cfzo 11540   |_cfl 11632   abscabs 12715    ~~> r crli 12955   O(1)co1 12956   sum_csu 13155   logclog 21981  Λcvma 22404  ψcchp 22405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-o1 12960  df-lo1 12961  df-sum 13156  df-ef 13345  df-e 13346  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-cxp 21984  df-cht 22409  df-vma 22410  df-chp 22411  df-ppi 22412
This theorem is referenced by:  selberg2  22775  selberg3lem2  22782
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