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Theorem selberg2lem 24467
Description: Lemma for selberg2 24468. 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 11331 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
2 chpcl 24130 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
31, 2syl 17 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
43recnd 9687 . . . . . . 7  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
5 rprege0 11339 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
6 flge0nn0 12087 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
75, 6syl 17 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e. 
NN0 )
8 nn0p1nn 10933 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  NN0  ->  ( ( |_ `  x )  +  1 )  e.  NN )
97, 8syl 17 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  NN )
109nnrpd 11362 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  RR+ )
1110relogcld 23651 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  RR )
1211recnd 9687 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  e.  CC )
13 relogcl 23604 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1413recnd 9687 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1512, 14subcld 10005 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
164, 15mulcld 9681 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  CC )
17 fzfid 12224 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
18 elfznn 11854 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1918adantl 473 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
2019nnrpd 11362 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
21 1rp 11329 . . . . . . . . . . . . 13  |-  1  e.  RR+
22 rpaddcl 11346 . . . . . . . . . . . . 13  |-  ( ( n  e.  RR+  /\  1  e.  RR+ )  ->  (
n  +  1 )  e.  RR+ )
2321, 22mpan2 685 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  +  1 )  e.  RR+ )
2423relogcld 23651 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  ( n  +  1 ) )  e.  RR )
25 relogcl 23604 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2624, 25resubcld 10068 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  e.  RR )
27 rpre 11331 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  RR )
28 chpcl 24130 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  (ψ `  n )  e.  RR )
2927, 28syl 17 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  RR )
3026, 29remulcld 9689 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  RR )
3130recnd 9687 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
3220, 31syl 17 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
3317, 32fsumcl 13876 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) )  e.  CC )
34 rpcnne0 11342 . . . . . 6  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
35 divsubdir 10325 . . . . . 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 1292 . . . . 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 9681 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  e.  CC )
384, 14mulcld 9681 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
3937, 38, 33sub32d 10037 . . . . . . 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 10095 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  (
( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) ) )
4140oveq1d 6323 . . . . . . 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 5879 . . . . . . . . . . 11  |-  ( m  =  n  ->  ( log `  m )  =  ( log `  n
) )
43 oveq1 6315 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
m  -  1 )  =  ( n  - 
1 ) )
4443fveq2d 5883 . . . . . . . . . . 11  |-  ( m  =  n  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) )
4542, 44jca 541 . . . . . . . . . 10  |-  ( m  =  n  ->  (
( log `  m
)  =  ( log `  n )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( n  -  1 ) ) ) )
46 fveq2 5879 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  ( log `  m )  =  ( log `  (
n  +  1 ) ) )
47 oveq1 6315 . . . . . . . . . . . 12  |-  ( m  =  ( n  + 
1 )  ->  (
m  -  1 )  =  ( ( n  +  1 )  - 
1 ) )
4847fveq2d 5883 . . . . . . . . . . 11  |-  ( m  =  ( n  + 
1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) )
4946, 48jca 541 . . . . . . . . . 10  |-  ( m  =  ( n  + 
1 )  ->  (
( log `  m
)  =  ( log `  ( n  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( n  + 
1 )  -  1 ) ) ) )
50 fveq2 5879 . . . . . . . . . . . 12  |-  ( m  =  1  ->  ( log `  m )  =  ( log `  1
) )
51 log1 23614 . . . . . . . . . . . 12  |-  ( log `  1 )  =  0
5250, 51syl6eq 2521 . . . . . . . . . . 11  |-  ( m  =  1  ->  ( log `  m )  =  0 )
53 oveq1 6315 . . . . . . . . . . . . . 14  |-  ( m  =  1  ->  (
m  -  1 )  =  ( 1  -  1 ) )
54 1m1e0 10700 . . . . . . . . . . . . . 14  |-  ( 1  -  1 )  =  0
5553, 54syl6eq 2521 . . . . . . . . . . . . 13  |-  ( m  =  1  ->  (
m  -  1 )  =  0 )
5655fveq2d 5883 . . . . . . . . . . . 12  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  0 ) )
57 2pos 10723 . . . . . . . . . . . . 13  |-  0  <  2
58 0re 9661 . . . . . . . . . . . . . 14  |-  0  e.  RR
59 chpeq0 24215 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR  ->  (
(ψ `  0 )  =  0  <->  0  <  2 ) )
6058, 59ax-mp 5 . . . . . . . . . . . . 13  |-  ( (ψ `  0 )  =  0  <->  0  <  2
)
6157, 60mpbir 214 . . . . . . . . . . . 12  |-  (ψ ` 
0 )  =  0
6256, 61syl6eq 2521 . . . . . . . . . . 11  |-  ( m  =  1  ->  (ψ `  ( m  -  1 ) )  =  0 )
6352, 62jca 541 . . . . . . . . . 10  |-  ( m  =  1  ->  (
( log `  m
)  =  0  /\  (ψ `  ( m  -  1 ) )  =  0 ) )
64 fveq2 5879 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  ( log `  m )  =  ( log `  (
( |_ `  x
)  +  1 ) ) )
65 oveq1 6315 . . . . . . . . . . . 12  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
m  -  1 )  =  ( ( ( |_ `  x )  +  1 )  - 
1 ) )
6665fveq2d 5883 . . . . . . . . . . 11  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )
6764, 66jca 541 . . . . . . . . . 10  |-  ( m  =  ( ( |_
`  x )  +  1 )  ->  (
( log `  m
)  =  ( log `  ( ( |_ `  x )  +  1 ) )  /\  (ψ `  ( m  -  1 ) )  =  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) ) )
68 nnuz 11218 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
699, 68syl6eleq 2559 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  e.  ( ZZ>= `  1 )
)
70 elfznn 11854 . . . . . . . . . . . . . 14  |-  ( m  e.  ( 1 ... ( ( |_ `  x )  +  1 ) )  ->  m  e.  NN )
7170adantl 473 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  NN )
7271nnrpd 11362 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR+ )
7372relogcld 23651 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  RR )
7473recnd 9687 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  ( log `  m )  e.  CC )
7571nnred 10646 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  m  e.  RR )
76 peano2rem 9961 . . . . . . . . . . . . 13  |-  ( m  e.  RR  ->  (
m  -  1 )  e.  RR )
7775, 76syl 17 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (
m  -  1 )  e.  RR )
78 chpcl 24130 . . . . . . . . . . . 12  |-  ( ( m  -  1 )  e.  RR  ->  (ψ `  ( m  -  1 ) )  e.  RR )
7977, 78syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  RR )
8079recnd 9687 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  m  e.  ( 1 ... (
( |_ `  x
)  +  1 ) ) )  ->  (ψ `  ( m  -  1 ) )  e.  CC )
8145, 49, 63, 67, 69, 74, 80fsumparts 13943 . . . . . . . . 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 11061 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  ZZ )
83 fzval3 12012 . . . . . . . . . . . 12  |-  ( ( |_ `  x )  e.  ZZ  ->  (
1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8482, 83syl 17 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  =  ( 1..^ ( ( |_ `  x )  +  1 ) ) )
8584eqcomd 2477 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1..^ ( ( |_ `  x )  +  1 ) )  =  ( 1 ... ( |_
`  x ) ) )
8619nncnd 10647 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  CC )
87 ax-1cn 9615 . . . . . . . . . . . . . . . . . 18  |-  1  e.  CC
88 pncan 9901 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  + 
1 )  -  1 )  =  n )
8986, 87, 88sylancl 675 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  n )
90 npcan 9904 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  +  1 )  =  n )
9186, 87, 90sylancl 675 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  -  1 )  +  1 )  =  n )
9289, 91eqtr4d 2508 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
n  +  1 )  -  1 )  =  ( ( n  - 
1 )  +  1 ) )
9392fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  ( ( n  - 
1 )  +  1 ) ) )
94 nnm1nn0 10935 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
9519, 94syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e. 
NN0 )
96 chpp1 24161 . . . . . . . . . . . . . . . 16  |-  ( ( n  -  1 )  e.  NN0  ->  (ψ `  ( ( n  - 
1 )  +  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  (
( n  -  1 )  +  1 ) ) ) )
9795, 96syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  -  1 )  +  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) ) )
9891fveq2d 5883 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  (
( n  -  1 )  +  1 ) )  =  (Λ `  n
) )
9998oveq2d 6324 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( n  -  1 ) )  +  (Λ `  ( ( n  - 
1 )  +  1 ) ) )  =  ( (ψ `  (
n  -  1 ) )  +  (Λ `  n
) ) )
10093, 97, 993eqtrd 2509 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  ( (ψ `  ( n  -  1 ) )  +  (Λ `  n ) ) )
101100oveq1d 6323 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  ( ( (ψ `  ( n  -  1
) )  +  (Λ `  n ) )  -  (ψ `  ( n  - 
1 ) ) ) )
10295nn0red 10950 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  -  1 )  e.  RR )
103 chpcl 24130 . . . . . . . . . . . . . . . 16  |-  ( ( n  -  1 )  e.  RR  ->  (ψ `  ( n  -  1 ) )  e.  RR )
104102, 103syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  RR )
105104recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
n  -  1 ) )  e.  CC )
106 vmacl 24124 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
10719, 106syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
108107recnd 9687 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
109105, 108pncan2d 10007 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(ψ `  ( n  -  1 ) )  +  (Λ `  n
) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
110101, 109eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (ψ `  ( ( n  + 
1 )  -  1 ) )  -  (ψ `  ( n  -  1 ) ) )  =  (Λ `  n )
)
111110oveq2d 6324 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  x.  ( (ψ `  (
( n  +  1 )  -  1 ) )  -  (ψ `  ( n  -  1
) ) ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
11220relogcld 23651 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
113112recnd 9687 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
114108, 113mulcomd 9682 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  =  ( ( log `  n )  x.  (Λ `  n ) ) )
115111, 114eqtr4d 2508 . . . . . . . . . 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 13850 . . . . . . . . 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 10951 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  CC )
118 pncan 9901 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  x
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( |_
`  x )  +  1 )  -  1 )  =  ( |_
`  x ) )
119117, 87, 118sylancl 675 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  -  1 )  =  ( |_ `  x
) )
120119fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  ( |_ `  x
) ) )
121 chpfl 24156 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR  ->  (ψ `  ( |_ `  x
) )  =  (ψ `  x ) )
1221, 121syl 17 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  (ψ `  ( |_ `  x ) )  =  (ψ `  x ) )
123120, 122eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) )  =  (ψ `  x ) )
124123oveq2d 6324 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  x ) ) )
12512, 4mulcomd 9682 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  x
) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
126124, 125eqtrd 2505 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  x.  (ψ `  (
( ( |_ `  x )  +  1 )  -  1 ) ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
127 0cn 9653 . . . . . . . . . . . . . 14  |-  0  e.  CC
128127mul01i 9841 . . . . . . . . . . . . 13  |-  ( 0  x.  0 )  =  0
129128a1i 11 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( 0  x.  0 )  =  0 )
130126, 129oveq12d 6326 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( ( (ψ `  x
)  x.  ( log `  ( ( |_ `  x )  +  1 ) ) )  - 
0 ) )
13137subid1d 9994 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) )  -  0 )  =  ( (ψ `  x )  x.  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
132130, 131eqtrd 2505 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( log `  (
( |_ `  x
)  +  1 ) )  x.  (ψ `  ( ( ( |_
`  x )  +  1 )  -  1 ) ) )  -  ( 0  x.  0 ) )  =  ( (ψ `  x )  x.  ( log `  (
( |_ `  x
)  +  1 ) ) ) )
13389fveq2d 5883 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
( n  +  1 )  -  1 ) )  =  (ψ `  n ) )
134133oveq2d 6324 . . . . . . . . . . 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 13850 . . . . . . . . . 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 6326 . . . . . . . . 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 2513 . . . . . . . 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 6323 . . . . . . 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 2515 . . . . . 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 6323 . . . . 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 10311 . . . . . . 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 1292 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  /  x )  =  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )
143142oveq1d 6323 . . . . 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 2514 . . . 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 4478 . . 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 6336 . . . . 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 6336 . . . . 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 9648 . . . . . . . 8  |-  RR  e.  _V
151 rpssre 11335 . . . . . . . 8  |-  RR+  C_  RR
152150, 151ssexi 4541 . . . . . . 7  |-  RR+  e.  _V
153152a1i 11 . . . . . 6  |-  ( T. 
->  RR+  e.  _V )
154 ovex 6336 . . . . . . 7  |-  ( (ψ `  x )  /  x
)  e.  _V
155154a1i 11 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  _V )
15615adantl 473 . . . . . 6  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  CC )
157 eqidd 2472 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  =  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) ) )
158 eqidd 2472 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )
159153, 155, 156, 157, 158offval2 6567 . . . . 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 24397 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
161 0red 9662 . . . . . . . 8  |-  ( T. 
->  0  e.  RR )
162 1red 9676 . . . . . . . 8  |-  ( T. 
->  1  e.  RR )
163 divrcnv 13987 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
x  e.  RR+  |->  ( 1  /  x ) )  ~~> r  0 )
16487, 163mp1i 13 . . . . . . . 8  |-  ( T. 
->  ( x  e.  RR+  |->  ( 1  /  x
) )  ~~> r  0 )
165 rpreccl 11349 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR+ )
166165rpred 11364 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1  /  x )  e.  RR )
167166adantl 473 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( 1  /  x )  e.  RR )
16811, 13resubcld 10068 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
169168adantl 473 . . . . . . . 8  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  e.  RR )
170 rpaddcl 11346 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  1  e.  RR+ )  ->  (
x  +  1 )  e.  RR+ )
17121, 170mpan2 685 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  +  1 )  e.  RR+ )
172171relogcld 23651 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( x  +  1 ) )  e.  RR )
173172, 13resubcld 10068 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  e.  RR )
1747nn0red 10950 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  e.  RR )
175 1red 9676 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  1  e.  RR )
176 flle 12068 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  ( |_ `  x )  <_  x )
1771, 176syl 17 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( |_
`  x )  <_  x )
178174, 1, 175, 177leadd1dd 10248 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( |_ `  x )  +  1 )  <_ 
( x  +  1 ) )
17910, 171logled 23655 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( ( |_ `  x
)  +  1 )  <_  ( x  + 
1 )  <->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) ) )
180178, 179mpbid 215 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( ( |_ `  x )  +  1 ) )  <_  ( log `  ( x  + 
1 ) ) )
18111, 172, 13, 180lesub1dd 10250 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
( log `  (
x  +  1 ) )  -  ( log `  x ) ) )
182 logdifbnd 23998 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( log `  ( x  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
183168, 173, 166, 181, 182letrd 9809 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) )  <_  (
1  /  x ) )
184183ad2antrl 742 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) )  <_ 
( 1  /  x
) )
185 fllep1 12070 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  x  <_  ( ( |_ `  x )  +  1 ) )
1861, 185syl 17 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  x  <_ 
( ( |_ `  x )  +  1 ) )
187 logleb 23631 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  (
( |_ `  x
)  +  1 )  e.  RR+ )  ->  (
x  <_  ( ( |_ `  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
18810, 187mpdan 681 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  <_  ( ( |_
`  x )  +  1 )  <->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) ) )
189186, 188mpbid 215 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  <_  ( log `  ( ( |_
`  x )  +  1 ) ) )
19011, 13subge0d 10224 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( 0  <_  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) )  <-> 
( log `  x
)  <_  ( log `  ( ( |_ `  x )  +  1 ) ) ) )
191189, 190mpbird 240 . . . . . . . . 9  |-  ( x  e.  RR+  ->  0  <_ 
( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )
192191ad2antrl 742 . . . . . . . 8  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
0  <_  ( ( log `  ( ( |_
`  x )  +  1 ) )  -  ( log `  x ) ) )
193161, 162, 164, 167, 169, 184, 192rlimsqz2 13791 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  ~~> r  0 )
194 rlimo1 13757 . . . . . . 7  |-  ( ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) )  ~~> r  0  ->  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) )  e.  O(1) )
195193, 194syl 17 . . . . . 6  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  (
( |_ `  x
)  +  1 ) )  -  ( log `  x ) ) )  e.  O(1) )
196 o1mul 13755 . . . . . 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 676 . . . . 5  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  oF  x.  ( x  e.  RR+  |->  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x ) ) ) )  e.  O(1) )
198159, 197eqeltrrd 2550 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( (ψ `  x )  /  x
)  x.  ( ( log `  ( ( |_ `  x )  +  1 ) )  -  ( log `  x
) ) ) )  e.  O(1) )
199 nnrp 11334 . . . . . . . . 9  |-  ( m  e.  NN  ->  m  e.  RR+ )
200199ssriv 3422 . . . . . . . 8  |-  NN  C_  RR+
201200a1i 11 . . . . . . 7  |-  ( T. 
->  NN  C_  RR+ )
202201sselda 3418 . . . . . 6  |-  ( ( T.  /\  n  e.  NN )  ->  n  e.  RR+ )
203202, 31syl 17 . . . . 5  |-  ( ( T.  /\  n  e.  NN )  ->  (
( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
204 chpo1ub 24397 . . . . . . . 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 11353 . . . . . . . . 9  |-  ( ( (ψ `  n )  e.  RR  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20729, 206mpancom 682 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  e.  RR )
208207adantl 473 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( (ψ `  n )  /  n
)  e.  RR )
20931adantl 473 . . . . . . 7  |-  ( ( T.  /\  n  e.  RR+ )  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  e.  CC )
210 rpreccl 11349 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR+ )
211210rpred 11364 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( 1  /  n )  e.  RR )
212 chpge0 24132 . . . . . . . . . . 11  |-  ( n  e.  RR  ->  0  <_  (ψ `  n )
)
21327, 212syl 17 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
(ψ `  n )
)
214 logdifbnd 23998 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( ( log `  ( n  +  1 ) )  -  ( log `  n
) )  <_  (
1  /  n ) )
21526, 211, 29, 213, 214lemul1ad 10568 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( ( ( log `  (
n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n )
)  <_  ( (
1  /  n )  x.  (ψ `  n
) ) )
21627lep1d 10560 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  n  <_ 
( n  +  1 ) )
217 logleb 23631 . . . . . . . . . . . . . 14  |-  ( ( n  e.  RR+  /\  (
n  +  1 )  e.  RR+ )  ->  (
n  <_  ( n  +  1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
21823, 217mpdan 681 . . . . . . . . . . . . 13  |-  ( n  e.  RR+  ->  ( n  <_  ( n  + 
1 )  <->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) ) )
219216, 218mpbid 215 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  <_  ( log `  ( n  + 
1 ) ) )
22024, 25subge0d 10224 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( 0  <_  ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  <-> 
( log `  n
)  <_  ( log `  ( n  +  1 ) ) ) )
221219, 220mpbird 240 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( ( log `  (
n  +  1 ) )  -  ( log `  n ) ) )
22226, 29, 221, 213mulge0d 10211 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  0  <_ 
( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
22330, 222absidd 13561 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  =  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )
224 rpregt0 11338 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( n  e.  RR  /\  0  <  n ) )
225 divge0 10496 . . . . . . . . . . . 12  |-  ( ( ( (ψ `  n
)  e.  RR  /\  0  <_  (ψ `  n
) )  /\  (
n  e.  RR  /\  0  <  n ) )  ->  0  <_  (
(ψ `  n )  /  n ) )
22629, 213, 224, 225syl21anc 1291 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  0  <_ 
( (ψ `  n
)  /  n ) )
227207, 226absidd 13561 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( (ψ `  n )  /  n
) )
22829recnd 9687 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  (ψ `  n )  e.  CC )
229 rpcn 11333 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  e.  CC )
230 rpne0 11340 . . . . . . . . . . 11  |-  ( n  e.  RR+  ->  n  =/=  0 )
231228, 229, 230divrec2d 10409 . . . . . . . . . 10  |-  ( n  e.  RR+  ->  ( (ψ `  n )  /  n
)  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
232227, 231eqtrd 2505 . . . . . . . . 9  |-  ( n  e.  RR+  ->  ( abs `  ( (ψ `  n
)  /  n ) )  =  ( ( 1  /  n )  x.  (ψ `  n
) ) )
233215, 223, 2323brtr4d 4426 . . . . . . . 8  |-  ( n  e.  RR+  ->  ( abs `  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  <_ 
( abs `  (
(ψ `  n )  /  n ) ) )
234233ad2antrl 742 . . . . . . 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 13793 . . . . . 6  |-  ( T. 
->  ( n  e.  RR+  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
236201, 235o1res2 13704 . . . . 5  |-  ( T. 
->  ( n  e.  NN  |->  ( ( ( log `  ( n  +  1 ) )  -  ( log `  n ) )  x.  (ψ `  n
) ) )  e.  O(1) )
237203, 236o1fsum 13950 . . . 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 13766 . . 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 2554 . 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 1461 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 189    /\ wa 376    = wceq 1452   T. wtru 1453    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810  ..^cfzo 11942   |_cfl 12059   abscabs 13374    ~~> r crli 13626   O(1)co1 13627   sum_csu 13829   logclog 23583  Λcvma 24097  ψcchp 24098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-o1 13631  df-lo1 13632  df-sum 13830  df-ef 14198  df-e 14199  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-cht 24102  df-vma 24103  df-chp 24104  df-ppi 24105
This theorem is referenced by:  selberg2  24468  selberg3lem2  24475
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