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Theorem vmadivsum 24399
Description: The sum of the von Mangoldt function over  n is asymptotic to  log x  +  O(1). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
Assertion
Ref Expression
vmadivsum  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( log `  x ) ) )  e.  O(1)
Distinct variable group:    x, n

Proof of Theorem vmadivsum
StepHypRef Expression
1 reex 9648 . . . . . . 7  |-  RR  e.  _V
2 rpssre 11335 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4541 . . . . . 6  |-  RR+  e.  _V
43a1i 11 . . . . 5  |-  ( T. 
->  RR+  e.  _V )
5 ovex 6336 . . . . . 6  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  e.  _V
65a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  e.  _V )
7 ovex 6336 . . . . . 6  |-  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  _V
87a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  e.  _V )
9 eqidd 2472 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) ) )
10 eqidd 2472 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )
114, 6, 8, 9, 10offval2 6567 . . . 4  |-  ( T. 
->  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  oF  -  ( x  e.  RR+  |->  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  =  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) ) )
1211trud 1461 . . 3  |-  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  oF  -  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  =  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )
13 fzfid 12224 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
14 elfznn 11854 . . . . . . . . . 10  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1514adantl 473 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
16 vmacl 24124 . . . . . . . . 9  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
1715, 16syl 17 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
1817, 15nndivred 10680 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  RR )
1913, 18fsumrecl 13877 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  e.  RR )
2019recnd 9687 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  e.  CC )
21 relogcl 23604 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2221recnd 9687 . . . . 5  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
23 rprege0 11339 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <_  x ) )
24 flge0nn0 12087 . . . . . . . . . 10  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( |_ `  x
)  e.  NN0 )
25 faccl 12507 . . . . . . . . . 10  |-  ( ( |_ `  x )  e.  NN0  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
2623, 24, 253syl 18 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ! `
 ( |_ `  x ) )  e.  NN )
2726nnrpd 11362 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ! `
 ( |_ `  x ) )  e.  RR+ )
2827relogcld 23651 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR )
29 rerpdivcl 11353 . . . . . . 7  |-  ( ( ( log `  ( ! `  ( |_ `  x ) ) )  e.  RR  /\  x  e.  RR+ )  ->  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x )  e.  RR )
3028, 29mpancom 682 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  RR )
3130recnd 9687 . . . . 5  |-  ( x  e.  RR+  ->  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x )  e.  CC )
3220, 22, 31nnncan2d 10040 . . . 4  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( log `  x ) ) )
3332mpteq2ia 4478 . . 3  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  -  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( log `  x ) ) )
3412, 33eqtri 2493 . 2  |-  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  oF  -  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( log `  x
) ) )
35 1red 9676 . . . . 5  |-  ( T. 
->  1  e.  RR )
36 chpo1ub 24397 . . . . . 6  |-  ( x  e.  RR+  |->  ( (ψ `  x )  /  x
) )  e.  O(1)
3736a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( (ψ `  x )  /  x ) )  e.  O(1) )
38 rpre 11331 . . . . . . . . 9  |-  ( x  e.  RR+  ->  x  e.  RR )
39 chpcl 24130 . . . . . . . . 9  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4038, 39syl 17 . . . . . . . 8  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
41 rerpdivcl 11353 . . . . . . . 8  |-  ( ( (ψ `  x )  e.  RR  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  RR )
4240, 41mpancom 682 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (ψ `  x )  /  x
)  e.  RR )
4342recnd 9687 . . . . . 6  |-  ( x  e.  RR+  ->  ( (ψ `  x )  /  x
)  e.  CC )
4443adantl 473 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( (ψ `  x )  /  x
)  e.  CC )
4520, 31subcld 10005 . . . . . 6  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  e.  CC )
4645adantl 473 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  e.  CC )
4738adantr 472 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  RR )
4818, 47remulcld 9689 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  n )  /  n )  x.  x
)  e.  RR )
49 nndivre 10667 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  n  e.  NN )  ->  ( x  /  n
)  e.  RR )
5038, 14, 49syl2an 485 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
51 reflcl 12065 . . . . . . . . . . . . 13  |-  ( ( x  /  n )  e.  RR  ->  ( |_ `  ( x  /  n ) )  e.  RR )
5250, 51syl 17 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( |_ `  ( x  /  n
) )  e.  RR )
5317, 52remulcld 9689 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( |_ `  ( x  /  n ) ) )  e.  RR )
5448, 53resubcld 10068 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( (Λ `  n )  /  n )  x.  x
)  -  ( (Λ `  n )  x.  ( |_ `  ( x  /  n ) ) ) )  e.  RR )
5550, 52resubcld 10068 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
x  /  n )  -  ( |_ `  ( x  /  n
) ) )  e.  RR )
56 1red 9676 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  1  e.  RR )
57 vmage0 24127 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  0  <_  (Λ `  n )
)
5815, 57syl 17 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  (Λ `  n ) )
59 fracle1 12072 . . . . . . . . . . . . 13  |-  ( ( x  /  n )  e.  RR  ->  (
( x  /  n
)  -  ( |_
`  ( x  /  n ) ) )  <_  1 )
6050, 59syl 17 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
x  /  n )  -  ( |_ `  ( x  /  n
) ) )  <_ 
1 )
6155, 56, 17, 58, 60lemul2ad 10569 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( x  /  n
)  -  ( |_
`  ( x  /  n ) ) ) )  <_  ( (Λ `  n )  x.  1 ) )
6217recnd 9687 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
6350recnd 9687 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  CC )
6452recnd 9687 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( |_ `  ( x  /  n
) )  e.  CC )
6562, 63, 64subdid 10095 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( x  /  n
)  -  ( |_
`  ( x  /  n ) ) ) )  =  ( ( (Λ `  n )  x.  ( x  /  n
) )  -  (
(Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
66 rpcn 11333 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR+  ->  x  e.  CC )
6766adantr 472 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  CC )
6815nnrpd 11362 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
69 rpcnne0 11342 . . . . . . . . . . . . . . 15  |-  ( n  e.  RR+  ->  ( n  e.  CC  /\  n  =/=  0 ) )
7068, 69syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  e.  CC  /\  n  =/=  0 ) )
71 div23 10311 . . . . . . . . . . . . . . 15  |-  ( ( (Λ `  n )  e.  CC  /\  x  e.  CC  /\  ( n  e.  CC  /\  n  =/=  0 ) )  -> 
( ( (Λ `  n
)  x.  x )  /  n )  =  ( ( (Λ `  n
)  /  n )  x.  x ) )
72 divass 10310 . . . . . . . . . . . . . . 15  |-  ( ( (Λ `  n )  e.  CC  /\  x  e.  CC  /\  ( n  e.  CC  /\  n  =/=  0 ) )  -> 
( ( (Λ `  n
)  x.  x )  /  n )  =  ( (Λ `  n
)  x.  ( x  /  n ) ) )
7371, 72eqtr3d 2507 . . . . . . . . . . . . . 14  |-  ( ( (Λ `  n )  e.  CC  /\  x  e.  CC  /\  ( n  e.  CC  /\  n  =/=  0 ) )  -> 
( ( (Λ `  n
)  /  n )  x.  x )  =  ( (Λ `  n
)  x.  ( x  /  n ) ) )
7462, 67, 70, 73syl3anc 1292 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  n )  /  n )  x.  x
)  =  ( (Λ `  n )  x.  (
x  /  n ) ) )
7574oveq1d 6323 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( (Λ `  n )  /  n )  x.  x
)  -  ( (Λ `  n )  x.  ( |_ `  ( x  /  n ) ) ) )  =  ( ( (Λ `  n )  x.  ( x  /  n
) )  -  (
(Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
7665, 75eqtr4d 2508 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( x  /  n
)  -  ( |_
`  ( x  /  n ) ) ) )  =  ( ( ( (Λ `  n
)  /  n )  x.  x )  -  ( (Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
7762mulid1d 9678 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  1 )  =  (Λ `  n
) )
7861, 76, 773brtr3d 4425 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( (Λ `  n )  /  n )  x.  x
)  -  ( (Λ `  n )  x.  ( |_ `  ( x  /  n ) ) ) )  <_  (Λ `  n
) )
7913, 54, 17, 78fsumle 13936 . . . . . . . . 9  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( (Λ `  n )  /  n
)  x.  x )  -  ( (Λ `  n
)  x.  ( |_
`  ( x  /  n ) ) ) )  <_  sum_ n  e.  ( 1 ... ( |_ `  x ) ) (Λ `  n )
)
8018recnd 9687 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  /  n
)  e.  CC )
8113, 66, 80fsummulc1 13923 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  /  n )  x.  x ) )
82 logfac2 24224 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  0  <_  x )  -> 
( log `  ( ! `  ( |_ `  x ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( |_
`  ( x  /  n ) ) ) )
8323, 82syl 17 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( |_
`  ( x  /  n ) ) ) )
8481, 83oveq12d 6326 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  =  (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  /  n )  x.  x )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
8548recnd 9687 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  n )  /  n )  x.  x
)  e.  CC )
8653recnd 9687 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( |_ `  ( x  /  n ) ) )  e.  CC )
8713, 85, 86fsumsub 13926 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( (Λ `  n )  /  n
)  x.  x )  -  ( (Λ `  n
)  x.  ( |_
`  ( x  /  n ) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  /  n )  x.  x
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( |_
`  ( x  /  n ) ) ) ) )
8884, 87eqtr4d 2508 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( (Λ `  n
)  /  n )  x.  x )  -  ( (Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
89 chpval 24128 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (ψ `  x )  =  sum_ n  e.  ( 1 ... ( |_ `  x
) ) (Λ `  n
) )
9038, 89syl 17 . . . . . . . . 9  |-  ( x  e.  RR+  ->  (ψ `  x )  =  sum_ n  e.  ( 1 ... ( |_ `  x
) ) (Λ `  n
) )
9179, 88, 903brtr4d 4426 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  <_  (ψ `  x ) )
9219, 38remulcld 9689 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  e.  RR )
9392, 28resubcld 10068 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  RR )
94 rpregt0 11338 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( x  e.  RR  /\  0  <  x ) )
95 lediv1 10492 . . . . . . . . 9  |-  ( ( ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  e.  RR  /\  (ψ `  x )  e.  RR  /\  ( x  e.  RR  /\  0  <  x ) )  -> 
( ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  <_  (ψ `  x
)  <->  ( ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x
)  <_  ( (ψ `  x )  /  x
) ) )
9693, 40, 94, 95syl3anc 1292 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  <_  (ψ `  x
)  <->  ( ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x
)  <_  ( (ψ `  x )  /  x
) ) )
9791, 96mpbid 215 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x )  <_  ( (ψ `  x )  /  x
) )
9892recnd 9687 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  e.  CC )
9928recnd 9687 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC )
100 rpcnne0 11342 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
101 divsubdir 10325 . . . . . . . . . . 11  |-  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  e.  CC  /\  ( log `  ( ! `  ( |_ `  x ) ) )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 ) )  ->  ( (
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x )  =  ( ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  /  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
10298, 99, 100, 101syl3anc 1292 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x )  =  ( ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  /  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
103 rpne0 11340 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  x  =/=  0 )
10420, 66, 103divcan4d 10411 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  /  x )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n ) )
105104oveq1d 6323 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  /  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )
106102, 105eqtr2d 2506 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) )  =  ( ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  x.  x
)  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x
) )
107106fveq2d 5883 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( abs `  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  =  ( abs `  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x ) ) )
108 rerpdivcl 11353 . . . . . . . . . 10  |-  ( ( ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  e.  RR  /\  x  e.  RR+ )  -> 
( ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x )  e.  RR )
10993, 108mpancom 682 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x )  e.  RR )
110 flle 12068 . . . . . . . . . . . . . . . 16  |-  ( ( x  /  n )  e.  RR  ->  ( |_ `  ( x  /  n ) )  <_ 
( x  /  n
) )
11150, 110syl 17 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( |_ `  ( x  /  n
) )  <_  (
x  /  n ) )
11250, 52subge0d 10224 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 0  <_  ( ( x  /  n )  -  ( |_ `  ( x  /  n ) ) )  <->  ( |_ `  ( x  /  n
) )  <_  (
x  /  n ) ) )
113111, 112mpbird 240 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( ( x  /  n
)  -  ( |_
`  ( x  /  n ) ) ) )
11417, 55, 58, 113mulge0d 10211 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( (Λ `  n )  x.  ( ( x  /  n )  -  ( |_ `  ( x  /  n ) ) ) ) )
115114, 76breqtrd 4420 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  0  <_  ( ( ( (Λ `  n
)  /  n )  x.  x )  -  ( (Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
11613, 54, 115fsumge0 13932 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  0  <_  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( (Λ `  n
)  /  n )  x.  x )  -  ( (Λ `  n )  x.  ( |_ `  (
x  /  n ) ) ) ) )
117116, 88breqtrrd 4422 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  0  <_ 
( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) ) )
118 divge0 10496 . . . . . . . . . 10  |-  ( ( ( ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  e.  RR  /\  0  <_  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) ) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x ) )
11993, 117, 94, 118syl21anc 1291 . . . . . . . . 9  |-  ( x  e.  RR+  ->  0  <_ 
( ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )
120109, 119absidd 13561 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( abs `  ( ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  x.  x )  -  ( log `  ( ! `  ( |_ `  x ) ) ) )  /  x ) )  =  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x ) )
121107, 120eqtrd 2505 . . . . . . 7  |-  ( x  e.  RR+  ->  ( abs `  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  =  ( ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  x.  x )  -  ( log `  ( ! `
 ( |_ `  x ) ) ) )  /  x ) )
122 chpge0 24132 . . . . . . . . . 10  |-  ( x  e.  RR  ->  0  <_  (ψ `  x )
)
12338, 122syl 17 . . . . . . . . 9  |-  ( x  e.  RR+  ->  0  <_ 
(ψ `  x )
)
124 divge0 10496 . . . . . . . . 9  |-  ( ( ( (ψ `  x
)  e.  RR  /\  0  <_  (ψ `  x
) )  /\  (
x  e.  RR  /\  0  <  x ) )  ->  0  <_  (
(ψ `  x )  /  x ) )
12540, 123, 94, 124syl21anc 1291 . . . . . . . 8  |-  ( x  e.  RR+  ->  0  <_ 
( (ψ `  x
)  /  x ) )
12642, 125absidd 13561 . . . . . . 7  |-  ( x  e.  RR+  ->  ( abs `  ( (ψ `  x
)  /  x ) )  =  ( (ψ `  x )  /  x
) )
12797, 121, 1263brtr4d 4426 . . . . . 6  |-  ( x  e.  RR+  ->  ( abs `  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  <_  ( abs `  ( (ψ `  x )  /  x
) ) )
128127ad2antrl 742 . . . . 5  |-  ( ( T.  /\  ( x  e.  RR+  /\  1  <_  x ) )  -> 
( abs `  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  /  n )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  <_  ( abs `  ( (ψ `  x
)  /  x ) ) )
12935, 37, 44, 46, 128o1le 13793 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  e.  O(1) )
130129trud 1461 . . 3  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  e.  O(1)
131 logfacrlim 24231 . . . 4  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1
132 rlimo1 13757 . . . 4  |-  ( ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  ~~> r  1  ->  ( x  e.  RR+  |->  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  e.  O(1) )
133131, 132ax-mp 5 . . 3  |-  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  e.  O(1)
134 o1sub 13756 . . 3  |-  ( ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  e.  O(1)  /\  ( x  e.  RR+  |->  ( ( log `  x
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  e.  O(1) )  -> 
( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  /  n )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) )  oF  -  ( x  e.  RR+  |->  ( ( log `  x )  -  (
( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  e.  O(1) )
135130, 133, 134mp2an 686 . 2  |-  ( ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( ( log `  ( ! `
 ( |_ `  x ) ) )  /  x ) ) )  oF  -  ( x  e.  RR+  |->  ( ( log `  x )  -  ( ( log `  ( ! `  ( |_ `  x ) ) )  /  x ) ) ) )  e.  O(1)
13634, 135eqeltrri 2546 1  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  /  n
)  -  ( log `  x ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   T. wtru 1453    e. wcel 1904    =/= wne 2641   _Vcvv 3031   class class class wbr 4395    |-> cmpt 4454   ` cfv 5589  (class class class)co 6308    oFcof 6548   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    x. cmul 9562    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   NNcn 10631   NN0cn0 10893   RR+crp 11325   ...cfz 11810   |_cfl 12059   !cfa 12497   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-cmp 20479  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:  vmadivsumb  24400  rpvmasumlem  24404  vmalogdivsum2  24455  vmalogdivsum  24456  2vmadivsumlem  24457  selberg3lem1  24474  selberg4lem1  24477  pntrsumo1  24482  selbergr  24485
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