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Theorem selberg2 21198
Description: Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg2  |-  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O
( 1 )
Distinct variable group:    x, n

Proof of Theorem selberg2
StepHypRef Expression
1 reex 9037 . . . . . . 7  |-  RR  e.  _V
2 rpssre 10578 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4308 . . . . . 6  |-  RR+  e.  _V
43a1i 11 . . . . 5  |-  (  T. 
->  RR+  e.  _V )
5 ovex 6065 . . . . . 6  |-  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  e.  _V
65a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  e.  _V )
7 ovex 6065 . . . . . 6  |-  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x )  e.  _V
87a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x )  e.  _V )
9 eqidd 2405 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) ) )  =  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) ) )
10 eqidd 2405 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  =  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) ) )
114, 6, 8, 9, 10offval2 6281 . . . 4  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) ) )
1211trud 1329 . . 3  |-  ( ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )
13 fzfid 11267 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
14 elfznn 11036 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1514adantl 453 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
16 vmacl 20854 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
1715, 16syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
1817recnd 9070 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  CC )
1915nnrpd 10603 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
20 relogcl 20426 . . . . . . . . . . . 12  |-  ( n  e.  RR+  ->  ( log `  n )  e.  RR )
2119, 20syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
2221recnd 9070 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  CC )
23 rpre 10574 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  RR )
24 nndivre 9991 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  n  e.  NN )  ->  ( x  /  n
)  e.  RR )
2523, 14, 24syl2an 464 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
26 chpcl 20860 . . . . . . . . . . . 12  |-  ( ( x  /  n )  e.  RR  ->  (ψ `  ( x  /  n
) )  e.  RR )
2725, 26syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  RR )
2827recnd 9070 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  CC )
2922, 28addcld 9063 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  n )  +  (ψ `  ( x  /  n ) ) )  e.  CC )
3018, 29mulcld 9064 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  e.  CC )
3113, 30fsumcl 12482 . . . . . . 7  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  e.  CC )
32 rpcn 10576 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
33 rpne0 10583 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
3431, 32, 33divcld 9746 . . . . . 6  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  e.  CC )
35 2cn 10026 . . . . . . 7  |-  2  e.  CC
36 relogcl 20426 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
3736recnd 9070 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
38 mulcl 9030 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( log `  x )  e.  CC )  -> 
( 2  x.  ( log `  x ) )  e.  CC )
3935, 37, 38sylancr 645 . . . . . 6  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  e.  CC )
4018, 22mulcld 9064 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  ( log `  n ) )  e.  CC )
4113, 40fsumcl 12482 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  e.  CC )
42 chpcl 20860 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4323, 42syl 16 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
4443recnd 9070 . . . . . . . . 9  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
4544, 37mulcld 9064 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
4641, 45subcld 9367 . . . . . . 7  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  e.  CC )
4746, 32, 33divcld 9746 . . . . . 6  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x )  e.  CC )
4834, 39, 47sub32d 9399 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) )  =  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  -  ( 2  x.  ( log `  x ) ) ) )
49 rpcnne0 10585 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
50 divsubdir 9666 . . . . . . . 8  |-  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  e.  CC  /\  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 ) )  -> 
( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  /  x )  =  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) ) )
5131, 46, 49, 50syl3anc 1184 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  /  x )  =  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) ) )
5218, 22, 28adddid 9068 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  =  ( ( (Λ `  n )  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) ) )
5352sumeq2dv 12452 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) ) )
5418, 28mulcld 9064 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
5513, 40, 54fsumadd 12487 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )
5653, 55eqtrd 2436 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )
5756oveq1d 6055 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  =  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) ) )
5813, 54fsumcl 12482 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
5941, 58, 45pnncand 9406 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  +  ( (ψ `  x
)  x.  ( log `  x ) ) ) )
6058, 45addcomd 9224 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  +  ( (ψ `  x
)  x.  ( log `  x ) ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) ) )
6157, 59, 603eqtrd 2440 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (
( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) ) )
6261oveq1d 6055 . . . . . . 7  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  -  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) ) )  /  x )  =  ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x ) )
6351, 62eqtr3d 2438 . . . . . 6  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  =  ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x ) )
6463oveq1d 6055 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  -  ( 2  x.  ( log `  x ) ) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )
6548, 64eqtrd 2436 . . . 4  |-  ( x  e.  RR+  ->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )
6665mpteq2ia 4251 . . 3  |-  ( x  e.  RR+  |->  ( ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) )  -  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )
6712, 66eqtri 2424 . 2  |-  ( ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )
68 selberg 21195 . . 3  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O
( 1 )
69 selberg2lem 21197 . . 3  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O ( 1 )
70 o1sub 12364 . . 3  |-  ( ( ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  (
x  /  n ) ) ) )  /  x )  -  (
2  x.  ( log `  x ) ) ) )  e.  O ( 1 )  /\  (
x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( log `  n
) )  -  (
(ψ `  x )  x.  ( log `  x
) ) )  /  x ) )  e.  O ( 1 ) )  ->  ( (
x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  e.  O
( 1 ) )
7168, 69, 70mp2an 654 . 2  |-  ( ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  ( ( log `  n
)  +  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  ( log `  n ) )  -  ( (ψ `  x )  x.  ( log `  x ) ) )  /  x ) ) )  e.  O
( 1 )
7267, 71eqeltrri 2475 1  |-  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O
( 1 )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    T. wtru 1322    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    e. cmpt 4226   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   RR+crp 10568   ...cfz 10999   |_cfl 11156   O ( 1 )co1 12235   sum_csu 12434   logclog 20405  Λcvma 20827  ψcchp 20828
This theorem is referenced by:  selberg2b  21199  selberg3  21206  selberg4  21208  selbergr  21215
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-o1 12239  df-lo1 12240  df-sum 12435  df-ef 12625  df-e 12626  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-em 20784  df-cht 20832  df-vma 20833  df-chp 20834  df-ppi 20835  df-mu 20836
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