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Theorem selberg3 22767
Description: Introduce a log weighting on the summands of  sum_ m  x.  n  <_  x , Λ ( m )Λ ( n ), the core of selberg2 22759 (written here as  sum_ n  <_  x , Λ ( n )ψ (
x  /  n )). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Assertion
Ref Expression
selberg3  |-  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1)
Distinct variable group:    x, n

Proof of Theorem selberg3
StepHypRef Expression
1 elioore 11326 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR )
21adantl 463 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  x  e.  RR )
3 chpcl 22421 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
42, 3syl 16 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (ψ `  x )  e.  RR )
5 1rp 10991 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
65a1i 11 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  1  e.  RR+ )
7 1red 9397 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  1  e.  RR )
8 eliooord 11351 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( 1 (,) +oo )  ->  ( 1  <  x  /\  x  < +oo ) )
98adantl 463 . . . . . . . . . . . . . . . 16  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
1  <  x  /\  x  < +oo ) )
109simpld 456 . . . . . . . . . . . . . . 15  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  1  <  x )
117, 2, 10ltled 9518 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  1  <_  x )
122, 6, 11rpgecld 11058 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  x  e.  RR+ )
1312relogcld 22031 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  ( log `  x )  e.  RR )
144, 13remulcld 9410 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(ψ `  x )  x.  ( log `  x
) )  e.  RR )
1514recnd 9408 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
(ψ `  x )  x.  ( log `  x
) )  e.  CC )
16 fzfid 11791 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
1 ... ( |_ `  x ) )  e. 
Fin )
17 elfznn 11474 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1817adantl 463 . . . . . . . . . . . . . 14  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
19 vmacl 22415 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
2018, 19syl 16 . . . . . . . . . . . . 13  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  n
)  e.  RR )
212adantr 462 . . . . . . . . . . . . . . 15  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  RR )
2221, 18nndivred 10366 . . . . . . . . . . . . . 14  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
23 chpcl 22421 . . . . . . . . . . . . . 14  |-  ( ( x  /  n )  e.  RR  ->  (ψ `  ( x  /  n
) )  e.  RR )
2422, 23syl 16 . . . . . . . . . . . . 13  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  n ) )  e.  RR )
2520, 24remulcld 9410 . . . . . . . . . . . 12  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  RR )
2616, 25fsumrecl 13207 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  e.  RR )
2726recnd 9408 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
28 2re 10387 . . . . . . . . . . . . . . 15  |-  2  e.  RR
2928a1i 11 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  2  e.  RR )
302, 10rplogcld 22037 . . . . . . . . . . . . . 14  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  ( log `  x )  e.  RR+ )
3129, 30rerpdivcld 11050 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
2  /  ( log `  x ) )  e.  RR )
3218nnrpd 11022 . . . . . . . . . . . . . . . 16  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3332relogcld 22031 . . . . . . . . . . . . . . 15  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
3425, 33remulcld 9410 . . . . . . . . . . . . . 14  |-  ( ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) )  e.  RR )
3516, 34fsumrecl 13207 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) )  e.  RR )
3631, 35remulcld 9410 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( 2  /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  e.  RR )
3736, 26resubcld 9772 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  RR )
3837recnd 9408 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  CC )
3915, 27, 38addassd 9404 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  +  ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )  =  ( ( (ψ `  x
)  x.  ( log `  x ) )  +  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  +  ( ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) ) ) )
40 2cnd 10390 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  2  e.  CC )
4113recnd 9408 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  ( log `  x )  e.  CC )
4230rpne0d 11028 . . . . . . . . . . . . 13  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  ( log `  x )  =/=  0 )
4340, 41, 42divcld 10103 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
2  /  ( log `  x ) )  e.  CC )
4435recnd 9408 . . . . . . . . . . . 12  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) )  e.  CC )
4543, 44mulcld 9402 . . . . . . . . . . 11  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( 2  /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  e.  CC )
4627, 45pncan3d 9718 . . . . . . . . . 10  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  +  ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )  =  ( ( 2  /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) ) )
4746oveq2d 6106 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  +  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  +  ( ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) ) )
4839, 47eqtr2d 2474 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  +  ( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  =  ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  +  ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) ) )
4948oveq1d 6105 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  +  ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )  /  x
) )
5014, 26readdcld 9409 . . . . . . . . 9  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  RR )
5150recnd 9408 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  CC )
522recnd 9408 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  x  e.  CC )
5312rpne0d 11028 . . . . . . . 8  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  x  =/=  0 )
5451, 38, 52, 53divdird 10141 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  +  ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) ) )  /  x
)  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) )
5549, 54eqtrd 2473 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) )
5655oveq1d 6105 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  =  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) )  -  ( 2  x.  ( log `  x
) ) ) )
5750, 12rerpdivcld 11050 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  e.  RR )
5857recnd 9408 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  e.  CC )
5937, 12rerpdivcld 11050 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x )  e.  RR )
6059recnd 9408 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x )  e.  CC )
6129, 13remulcld 9410 . . . . . . 7  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
2  x.  ( log `  x ) )  e.  RR )
6261recnd 9408 . . . . . 6  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
2  x.  ( log `  x ) )  e.  CC )
6358, 60, 62addsubd 9736 . . . . 5  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  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
) ) )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) )
6456, 63eqtrd 2473 . . . 4  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  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 ) ) )  +  ( ( ( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) )
6564mpteq2dva 4375 . . 3  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  ( ( 2  /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  =  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  +  ( ( ( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) ) )
6657, 61resubcld 9772 . . . 4  |-  ( ( T.  /\  x  e.  ( 1 (,) +oo ) )  ->  (
( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  e.  RR )
6712ex 434 . . . . . 6  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  ->  x  e.  RR+ )
)
6867ssrdv 3359 . . . . 5  |-  ( T. 
->  ( 1 (,) +oo )  C_  RR+ )
69 selberg2 22759 . . . . . 6  |-  ( 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)
7069a1i 11 . . . . 5  |-  ( T. 
->  ( 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) )
7168, 70o1res2 13037 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1) )
72 selberg3lem2 22766 . . . . 5  |-  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( 2  / 
( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) )  e.  O(1)
7372a1i 11 . . . 4  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) )  e.  O(1) )
7466, 59, 71, 73o1add2 13097 . . 3  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x )  -  ( 2  x.  ( log `  x
) ) )  +  ( ( ( ( 2  /  ( log `  x ) )  x. 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  /  x ) ) )  e.  O(1) )
7565, 74eqeltrd 2515 . 2  |-  ( T. 
->  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  ( ( 2  /  ( log `  x ) )  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1) )
7675trud 1373 1  |-  ( x  e.  ( 1 (,) +oo )  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  ( ( 2  /  ( log `  x
) )  x.  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369   T. wtru 1365    e. wcel 1761   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   RRcr 9277   1c1 9279    + caddc 9281    x. cmul 9283   +oocpnf 9411    < clt 9414    - cmin 9591    / cdiv 9989   NNcn 10318   2c2 10367   RR+crp 10987   (,)cioo 11296   ...cfz 11433   |_cfl 11636   O(1)co1 12960   sum_csu 13159   logclog 21965  Λcvma 22388  ψcchp 22389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-o1 12964  df-lo1 12965  df-sum 13160  df-ef 13349  df-e 13350  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-fbas 17773  df-fg 17774  df-cnfld 17778  df-top 18462  df-bases 18464  df-topon 18465  df-topsp 18466  df-cld 18582  df-ntr 18583  df-cls 18584  df-nei 18661  df-lp 18699  df-perf 18700  df-cn 18790  df-cnp 18791  df-haus 18878  df-cmp 18949  df-tx 19094  df-hmeo 19287  df-fil 19378  df-fm 19470  df-flim 19471  df-flf 19472  df-xms 19854  df-ms 19855  df-tms 19856  df-cncf 20413  df-limc 21300  df-dv 21301  df-log 21967  df-cxp 21968  df-em 22345  df-cht 22393  df-vma 22394  df-chp 22395  df-ppi 22396  df-mu 22397
This theorem is referenced by:  selberg3r  22777
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