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Theorem selberg2lem 21111
Description: Lemma for selberg2 21112. 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 10550 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
2 chpcl 20774 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
31, 2syl 16 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
43recnd 9047 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
5 rprege0 10558 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
6 flge0nn0 11152 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
75, 6syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e. 
NN0 )
8 nn0p1nn 10191 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  NN0  ->  ( ( |_ `  x )  +  1 )  e.  NN )
97, 8syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  NN )
109nnrpd 10579 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  RR+ )
1110relogcld 20385 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  RR )
1211recnd 9047 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  CC )
13 relogcl 20340 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1413recnd 9047 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1512, 14subcld 9343 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
164, 15mulcld 9041 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  CC )
17 fzfid 11239 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
18 elfznn 11012 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1918adantl 453 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
2019nnrpd 10579 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
21 1rp 10548 . . . . . . . . . . . . 13  |-  1  e.  RR+
22 rpaddcl 10564 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR+  /\  1  e.  RR+ )  ->  (
n  +  1 )  e.  RR+ )
2321, 22mpan2 653 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  +  1 )  e.  RR+ )
2423relogcld 20385 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  ( n  +  1 ) )  e.  RR )
25 relogcl 20340 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2624, 25resubcld 9397 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  e.  RR )
27 rpre 10550 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
28 chpcl 20774 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  (ψ `  n )  e.  RR )
2927, 28syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  RR )
3026, 29remulcld 9049 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  RR )
3130recnd 9047 . . . . . . . 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 12454 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC )
34 rpcnne0 10561 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
35 divsubdir 9642 . . . . . 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 1184 . . . . 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 9041 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  e.  CC )
384, 14mulcld 9041 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
3937, 38, 33sub32d 9375 . . . . . . 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 9421 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) ) )
4140oveq1d 6035 . . . . . . 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 5668 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( log `  m )  =  ( log `  n
) )
43 oveq1 6027 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  -  1 )  =  ( n  - 
1 ) )
4443fveq2d 5672 . . . . . . . . . . 11  |-  ( m  =  n  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) )
4542, 44jca 519 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( log `  m
)  =  ( log `  n )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) ) )
46 fveq2 5668 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  ( log `  m )  =  ( log `  (
n  +  1 ) ) )
47 oveq1 6027 . . . . . . . . . . . 12  |-  ( m  =  ( n  + 
1 )  ->  (
m  -  1 )  =  ( ( n  +  1 )  - 
1 ) )
4847fveq2d 5672 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) )
4946, 48jca 519 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
( log `  m
)  =  ( log `  ( n  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) )
50 fveq2 5668 . . . . . . . . . . . 12  |-  ( m  =  1  ->  ( log `  m )  =  ( log `  1
) )
51 log1 20347 . . . . . . . . . . . 12  |-  ( log `  1 )  =  0
5250, 51syl6eq 2435 . . . . . . . . . . 11  |-  ( m  =  1  ->  ( log `  m )  =  0 )
53 oveq1 6027 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
m  -  1 )  =  ( 1  -  1 ) )
54 1m1e0 10000 . . . . . . . . . . . . . 14  |-  ( 1  -  1 )  =  0
5553, 54syl6eq 2435 . . . . . . . . . . . . 13  |-  ( m  =  1  ->  (
m  -  1 )  =  0 )
5655fveq2d 5672 . . . . . . . . . . . 12  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  0 ) )
57 2pos 10014 . . . . . . . . . . . . 13  |-  0  <  2
58 0re 9024 . . . . . . . . . . . . . 14  |-  0  e.  RR
59 chpeq0 20859 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  (
(ψ `  0 )  =  0  <->  0  <  2 ) )
6058, 59ax-mp 8 . . . . . . . . . . . . 13  |-  ( (ψ `  0 )  =  0  <->  0  <  2
)
6157, 60mpbir 201 . . . . . . . . . . . 12  |-  (ψ ` 
0 )  =  0
6256, 61syl6eq 2435 . . . . . . . . . . 11  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  0 )
6352, 62jca 519 . . . . . . . . . 10  |-  ( m  =  1  ->  (
( log `  m
)  =  0  /\  (ψ `  ( m  -  1 ) )  =  0 ) )
64 fveq2 5668 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  ( log `  m )  =  ( log `  (
( |_ `  x
)  +  1 ) ) )
65 oveq1 6027 . . . . . . . . . . . 12  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
m  -  1 )  =  ( ( ( |_ `  x )  +  1 )  - 
1 ) )
6665fveq2d 5672 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )
6764, 66jca 519 . . . . . . . . . 10  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
( log `  m
)  =  ( log `  ( ( |_ `  x )  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) ) )
68 nnuz 10453 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
699, 68syl6eleq 2477 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  ( ZZ>= `  1 )
)
70 elfznn 11012 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... ( ( |_ `  x )  +  1 ) )  ->  m  e.  NN )
7170adantl 453 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  NN )
7271nnrpd 10579 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR+ )
7372relogcld 20385 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  RR )
7473recnd 9047 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  CC )
7571nnred 9947 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR )
76 peano2rem 9299 . . . . . . . . . . . . 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 20774 . . . . . . . . . . . 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 9047 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  CC )
8145, 49, 63, 67, 69, 74, 80fsumparts 12512 . . . . . . . . 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 10305 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  ZZ )
83 fzval3 11108 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  ZZ  ->  (
1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8482, 83syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8584eqcomd 2392 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1..^ ( ( |_ `  x )  +  1 ) )  =  ( 1 ... ( |_
`  x ) ) )
8619nncnd 9948 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  CC )
87 ax-1cn 8981 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
88 pncan 9243 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  + 
1 )  -  1 )  =  n )
8986, 87, 88sylancl 644 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  n )
90 npcan 9246 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
9186, 87, 90sylancl 644 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  -  1 )  +  1 )  =  n )
9289, 91eqtr4d 2422 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  ( ( n  - 
1 )  +  1 ) )
9392fveq2d 5672 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  ( ( n  - 
1 )  +  1 ) ) )
94 nnm1nn0 10193 . . . . . . . . . . . . . . . . 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 20805 . . . . . . . . . . . . . . . 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 5672 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  (
( n  -  1 )  +  1 ) )  =  (Λ `  n
) )
9998oveq2d 6036 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) )  =  ( (ψ `  (
n  -  1 ) )  +  (Λ `  n
) ) )
10093, 97, 993eqtrd 2423 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  n ) ) )
101100oveq1d 6035 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  ( ( (ψ `  ( n  -  1
) )  +  (Λ `  n ) )  -  (ψ `  ( n  - 
1 ) ) ) )
10295nn0red 10207 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e.  RR )
103 chpcl 20774 . . . . . . . . . . . . . . . 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 9047 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  CC )
106 vmacl 20768 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
10719, 106syl 16 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
108107recnd 9047 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
109105, 108pncan2d 9345 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(ψ `  ( n  -  1 ) )  +  (Λ `  n
) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
110101, 109eqtrd 2419 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
111110oveq2d 6036 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
11220relogcld 20385 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
113112recnd 9047 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
114108, 113mulcomd 9042 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
115111, 114eqtr4d 2422 . . . . . . . . . 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 12428 . . . . . . . . 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 10208 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  CC )
118 pncan 9243 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( |_
`  x )  +  1 )  -  1 )  =  ( |_
`  x ) )
119117, 87, 118sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  -  1 )  =  ( |_ `  x
) )
120119fveq2d 5672 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  ( |_ `  x
) ) )
121 chpfl 20800 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR  ->  (ψ `  ( |_ `  x
) )  =  (ψ `  x ) )
1221, 121syl 16 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( |_ `  x ) )  =  (ψ `  x ) )
123120, 122eqtrd 2419 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  x ) )
124123oveq2d 6036 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  x ) ) )
12512, 4mulcomd 9042 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  x
) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
126124, 125eqtrd 2419 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
127 0cn 9017 . . . . . . . . . . . . . 14  |-  0  e.  CC
128127mul01i 9188 . . . . . . . . . . . . 13  |-  ( 0  x.  0 )  =  0
129128a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 0  x.  0 )  =  0 )
130126, 129oveq12d 6038 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  - 
0 ) )
13137subid1d 9332 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) )  -  0 )  =  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
132130, 131eqtrd 2419 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
13389fveq2d 5672 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  n ) )
134133oveq2d 6036 . . . . . . . . . . 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 12428 . . . . . . . . . 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 6038 . . . . . . . . 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 2427 . . . . . . . 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 6035 . . . . . . 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 2429 . . . . . 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 6035 . . . . 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 9629 . . . . . . 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 1184 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
143142oveq1d 6035 . . . . 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 2428 . . . 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 4232 . . 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 6045 . . . . 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 6045 . . . . 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 9014 . . . . . . . 8  |-  RR  e.  _V
151 rpssre 10554 . . . . . . . 8  |-  RR+  C_  RR
152150, 151ssexi 4289 . . . . . . 7  |-  RR+  e.  _V
153152a1i 11 . . . . . 6  |-  (  T. 
->  RR+  e.  _V )
154 ovex 6045 . . . . . . 7  |-  ( (ψ `  x )  /  x
)  e.  _V
155154a1i 11 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  _V )
15615adantl 453 . . . . . 6  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
157 eqidd 2388 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) ) )
158 eqidd 2388 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )
159153, 155, 156, 157, 158offval2 6261 . . . . 5  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  o F  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  =  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) ) ) )
160 chpo1ub 21041 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O
( 1 )
16158a1i 11 . . . . . . . 8  |-  (  T. 
->  0  e.  RR )
162 1re 9023 . . . . . . . . 9  |-  1  e.  RR
163162a1i 11 . . . . . . . 8  |-  (  T. 
->  1  e.  RR )
164 divrcnv 12559 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
16587, 164mp1i 12 . . . . . . . 8  |-  (  T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
166 rpreccl 10567 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
167166rpred 10580 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR )
168167adantl 453 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
16911, 13resubcld 9397 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
170169adantl 453 . . . . . . . 8  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
171 rpaddcl 10564 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
17221, 171mpan2 653 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  +  1 )  e.  RR+ )
173172relogcld 20385 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( x  +  1 ) )  e.  RR )
174173, 13resubcld 9397 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  e.  RR )
1757nn0red 10207 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  RR )
176162a1i 11 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  1  e.  RR )
177 flle 11135 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( |_ `  x )  <_  x )
1781, 177syl 16 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  <_  x )
179175, 1, 176, 178leadd1dd 9572 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  <_ 
( x  +  1 ) )
18010, 172logled 20389 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  <_  ( x  + 
1 )  <->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) ) )
181179, 180mpbid 202 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) )
18211, 173, 13, 181lesub1dd 9574 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
( log `  (
x  +  1 ) )  -  ( log `  x ) ) )
183 logdifbnd 20699 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
184169, 174, 167, 182, 183letrd 9159 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
185184ad2antrl 709 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) )  <_ 
( 1  /  x
) )
186 fllep1 11137 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <_  ( ( |_ `  x )  +  1 ) )
1871, 186syl 16 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  <_ 
( ( |_ `  x )  +  1 ) )
188 logleb 20365 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  (
( |_ `  x
)  +  1 )  e.  RR+ )  ->  (
x  <_  ( ( |_ `  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
18910, 188mpdan 650 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  <_  ( ( |_
`  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
190187, 189mpbid 202 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) )
19111, 13subge0d 9548 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 0  <_  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) )  <-> 
( log `  x
)  <_  ( log `  ( ( |_ `  x )  +  1 ) ) ) )
192190, 191mpbird 224 . . . . . . . . 9  |-  ( x  e.  RR+  ->  0  <_ 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )
193192ad2antrl 709 . . . . . . . 8  |-  ( (  T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( ( log `  ( ( |_
`  x )  +  1 ) )  -  ( log `  x ) ) )
194161, 163, 165, 168, 170, 185, 193rlimsqz2 12371 . . . . . . 7  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  ~~> r  0 )
195 rlimo1 12337 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  ~~> r  0  ->  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
196194, 195syl 16 . . . . . 6  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  O ( 1 ) )
197 o1mul 12335 . . . . . 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
) )  o F  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O
( 1 ) )
198160, 196, 197sylancr 645 . . . . 5  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  o F  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O
( 1 ) )
199159, 198eqeltrrd 2462 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )  e.  O ( 1 ) )
200 nnrp 10553 . . . . . . . . 9  |-  ( m  e.  NN  ->  m  e.  RR+ )
201200ssriv 3295 . . . . . . . 8  |-  NN  C_  RR+
202201a1i 11 . . . . . . 7  |-  (  T. 
->  NN  C_  RR+ )
203202sselda 3291 . . . . . 6  |-  ( (  T.  /\  n  e.  NN )  ->  n  e.  RR+ )
204203, 31syl 16 . . . . 5  |-  ( (  T.  /\  n  e.  NN )  ->  (
( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
205 chpo1ub 21041 . . . . . . . 8  |-  ( n  e.  RR+  |->  ( (ψ `  n )  /  n
) )  e.  O
( 1 )
206205a1i 11 . . . . . . 7  |-  (  T. 
->  ( n  e.  RR+  |->  ( (ψ `  n )  /  n ) )  e.  O ( 1 ) )
207 rerpdivcl 10571 . . . . . . . . 9  |-  ( ( (ψ `  n )  e.  RR  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20829, 207mpancom 651 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  e.  RR )
209208adantl 453 . . . . . . 7  |-  ( (  T.  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
21031adantl 453 . . . . . . 7  |-  ( (  T.  /\  n  e.  RR+ )  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
211 rpreccl 10567 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR+ )
212211rpred 10580 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR )
213 chpge0 20776 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  0  <_  (ψ `  n )
)
21427, 213syl 16 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
(ψ `  n )
)
215 logdifbnd 20699 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  <_  (
1  /  n ) )
21626, 212, 29, 214, 215lemul1ad 9882 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  <_  ( (
1  /  n )  x.  (ψ `  n
) ) )
21727lep1d 9874 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  n  <_ 
( n  +  1 ) )
218 logleb 20365 . . . . . . . . . . . . . 14  |-  ( ( n  e.  RR+  /\  (
n  +  1 )  e.  RR+ )  ->  (
n  <_  ( n  +  1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
21923, 218mpdan 650 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  ( n  <_  ( n  + 
1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
220217, 219mpbid 202 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) )
22124, 25subge0d 9548 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( 0  <_  ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  <-> 
( log `  n
)  <_  ( log `  ( n  +  1 ) ) ) )
222220, 221mpbird 224 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( ( log `  (
n  +  1 ) )  -  ( log `  n ) ) )
22326, 29, 222, 214mulge0d 9535 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
22430, 223absidd 12152 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
225 rpregt0 10557 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  e.  RR  /\  0  <  n ) )
226 divge0 9811 . . . . . . . . . . . 12  |-  ( ( ( (ψ `  n
)  e.  RR  /\  0  <_  (ψ `  n
) )  /\  (
n  e.  RR  /\  0  <  n ) )  ->  0  <_  (
(ψ `  n )  /  n ) )
22729, 214, 225, 226syl21anc 1183 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( (ψ `  n
)  /  n ) )
228208, 227absidd 12152 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( (ψ `  n )  /  n
) )
22929recnd 9047 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  CC )
230 rpcn 10552 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  CC )
231 rpne0 10559 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  =/=  0 )
232229, 230, 231divrec2d 9726 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
233228, 232eqtrd 2419 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
234216, 224, 2333brtr4d 4183 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  <_ 
( abs `  (
(ψ `  n )  /  n ) ) )
235234ad2antrl 709 . . . . . . 7  |-  ( (  T.  /\  ( n  e.  RR+  /\  1  <_  n ) )  -> 
( abs `  (
( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
) )  <_  ( abs `  ( (ψ `  n )  /  n
) ) )
236163, 206, 209, 210, 235o1le 12373 . . . . . 6  |-  (  T. 
->  ( n  e.  RR+  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O ( 1 ) )
237202, 236o1res2 12284 . . . . 5  |-  (  T. 
->  ( n  e.  NN  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O ( 1 ) )
238204, 237o1fsum 12519 . . . 4  |-  (  T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  /  x
) )  e.  O
( 1 ) )
239147, 149, 199, 238o1sub2 12346 . . 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 ) )
240145, 239syl5eqelr 2472 . 2  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O ( 1 ) )
241240trud 1329 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 set class
Syntax hints:    <-> wb 177    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    C_ wss 3263   class class class wbr 4153    e. cmpt 4207   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   NNcn 9932   2c2 9981   NN0cn0 10153   ZZcz 10214   ZZ>=cuz 10420   RR+crp 10544   ...cfz 10975  ..^cfzo 11065   |_cfl 11128   abscabs 11966    ~~> r crli 12206   O (
1 )co1 12207   sum_csu 12406   logclog 20319  Λcvma 20741  ψcchp 20742
This theorem is referenced by:  selberg2  21112  selberg3lem2  21119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ioc 10853  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-mod 11178  df-seq 11251  df-exp 11310  df-fac 11494  df-bc 11521  df-hash 11546  df-shft 11809  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-limsup 12192  df-clim 12209  df-rlim 12210  df-o1 12211  df-lo1 12212  df-sum 12407  df-ef 12597  df-e 12598  df-sin 12599  df-cos 12600  df-pi 12602  df-dvds 12780  df-gcd 12934  df-prm 13007  df-pc 13138  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-fbas 16623  df-fg 16624  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123  df-perf 17124  df-cn 17213  df-cnp 17214  df-haus 17301  df-tx 17515  df-hmeo 17708  df-fil 17799  df-fm 17891  df-flim 17892  df-flf 17893  df-xms 18259  df-ms 18260  df-tms 18261  df-cncf 18779  df-limc 19620  df-dv 19621  df-log 20321  df-cxp 20322  df-cht 20746  df-vma 20747  df-chp 20748  df-ppi 20749
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