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Theorem selberg3 21258
Description: Introduce a log weighting on the summands of  sum_ m  x.  n  <_  x , Λ ( m )Λ ( n ), the core of selberg2 21250 (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 10951 . . . . . . . . . . . . . 14  |-  ( x  e.  ( 1 (,) 
+oo )  ->  x  e.  RR )
21adantl 454 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR )
3 chpcl 20912 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
42, 3syl 16 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (ψ `  x )  e.  RR )
5 1rp 10621 . . . . . . . . . . . . . . 15  |-  1  e.  RR+
65a1i 11 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR+ )
76rpred 10653 . . . . . . . . . . . . . . 15  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  e.  RR )
8 eliooord 10975 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( 1 (,) 
+oo )  ->  (
1  <  x  /\  x  <  +oo ) )
98adantl 454 . . . . . . . . . . . . . . . 16  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1  <  x  /\  x  <  +oo ) )
109simpld 447 . . . . . . . . . . . . . . 15  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <  x )
117, 2, 10ltled 9226 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  1  <_  x )
122, 6, 11rpgecld 10688 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  RR+ )
1312relogcld 20523 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( log `  x )  e.  RR )
144, 13remulcld 9121 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
(ψ `  x )  x.  ( log `  x
) )  e.  RR )
1514recnd 9119 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
(ψ `  x )  x.  ( log `  x
) )  e.  CC )
16 fzfid 11317 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
1 ... ( |_ `  x ) )  e. 
Fin )
17 elfznn 11085 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
1817adantl 454 . . . . . . . . . . . . . 14  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
19 vmacl 20906 . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . . 15  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  RR )
2221, 18nndivred 10053 . . . . . . . . . . . . . 14  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR )
23 chpcl 20912 . . . . . . . . . . . . . 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 9121 . . . . . . . . . . . 12  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  n )  x.  (ψ `  ( x  /  n
) ) )  e.  RR )
2616, 25fsumrecl 12533 . . . . . . . . . . 11  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  e.  RR )
2726recnd 9119 . . . . . . . . . 10  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  e.  CC )
28 2re 10074 . . . . . . . . . . . . . . 15  |-  2  e.  RR
2928a1i 11 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  2  e.  RR )
302, 10rplogcld 20529 . . . . . . . . . . . . . 14  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( log `  x )  e.  RR+ )
3129, 30rerpdivcld 10680 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
2  /  ( log `  x ) )  e.  RR )
3218nnrpd 10652 . . . . . . . . . . . . . . . 16  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3332relogcld 20523 . . . . . . . . . . . . . . 15  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  n )  e.  RR )
3425, 33remulcld 9121 . . . . . . . . . . . . . 14  |-  ( ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  n )  x.  (ψ `  ( x  /  n ) ) )  x.  ( log `  n
) )  e.  RR )
3516, 34fsumrecl 12533 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) )  e.  RR )
3631, 35remulcld 9121 . . . . . . . . . . . 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 9470 . . . . . . . . . . 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 9119 . . . . . . . . . 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 9115 . . . . . . . . 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 2cn 10075 . . . . . . . . . . . . . 14  |-  2  e.  CC
4140a1i 11 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  2  e.  CC )
4213recnd 9119 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( log `  x )  e.  CC )
4330rpne0d 10658 . . . . . . . . . . . . 13  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  ( log `  x )  =/=  0 )
4441, 42, 43divcld 9795 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
2  /  ( log `  x ) )  e.  CC )
4535recnd 9119 . . . . . . . . . . . 12  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  n
)  x.  (ψ `  ( x  /  n
) ) )  x.  ( log `  n
) )  e.  CC )
4644, 45mulcld 9113 . . . . . . . . . . 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 )
4727, 46pncan3d 9419 . . . . . . . . . 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
) ) ) )
4847oveq2d 6100 . . . . . . . . 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
) ) ) ) )
4939, 48eqtr2d 2471 . . . . . . . 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 ) ) ) ) ) )
5049oveq1d 6099 . . . . . . 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
) )
5114, 26readdcld 9120 . . . . . . . . 9  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  RR )
5251recnd 9119 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( (Λ `  n )  x.  (ψ `  ( x  /  n ) ) ) )  e.  CC )
532recnd 9119 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  e.  CC )
5412rpne0d 10658 . . . . . . . 8  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  x  =/=  0 )
5552, 38, 53, 54divdird 9833 . . . . . . 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 ) ) )
5650, 55eqtrd 2470 . . . . . 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 ) ) )
5756oveq1d 6099 . . . . 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
) ) ) )
5851, 12rerpdivcld 10680 . . . . . . 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 )
5958recnd 9119 . . . . . 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 )
6037, 12rerpdivcld 10680 . . . . . . 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 )
6160recnd 9119 . . . . . 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 )
6229, 13remulcld 9121 . . . . . . 7  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
2  x.  ( log `  x ) )  e.  RR )
6362recnd 9119 . . . . . 6  |-  ( (  T.  /\  x  e.  ( 1 (,)  +oo ) )  ->  (
2  x.  ( log `  x ) )  e.  CC )
6459, 61, 63addsubd 9437 . . . . 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 ) ) )
6557, 64eqtrd 2470 . . . 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 ) ) )
6665mpteq2dva 4298 . . 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 ) ) ) )
6758, 62resubcld 9470 . . . 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 )
6812ex 425 . . . . . 6  |-  (  T. 
->  ( x  e.  ( 1 (,)  +oo )  ->  x  e.  RR+ )
)
6968ssrdv 3356 . . . . 5  |-  (  T. 
->  ( 1 (,)  +oo )  C_  RR+ )
70 selberg2 21250 . . . . . 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 )
7170a1i 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 ) )
7269, 71o1res2 12362 . . . 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 ) )
73 selberg3lem2 21257 . . . . 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 )
7473a1i 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 ) )
7567, 60, 72, 74o1add2 12422 . . 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 ) )
7666, 75eqeltrd 2512 . 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 ) )
7776trud 1333 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 set class
Syntax hints:    /\ wa 360    T. wtru 1326    e. wcel 1726   class class class wbr 4215    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   CCcc 8993   RRcr 8994   1c1 8996    + caddc 8998    x. cmul 9000    +oocpnf 9122    < clt 9125    - cmin 9296    / cdiv 9682   NNcn 10005   2c2 10054   RR+crp 10617   (,)cioo 10921   ...cfz 11048   |_cfl 11206   O ( 1 )co1 12285   sum_csu 12484   logclog 20457  Λcvma 20879  ψcchp 20880
This theorem is referenced by:  selberg3r  21268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074  ax-mulf 9075
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-ixp 7067  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ioo 10925  df-ioc 10926  df-ico 10927  df-icc 10928  df-fz 11049  df-fzo 11141  df-fl 11207  df-mod 11256  df-seq 11329  df-exp 11388  df-fac 11572  df-bc 11599  df-hash 11624  df-shft 11887  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-limsup 12270  df-clim 12287  df-rlim 12288  df-o1 12289  df-lo1 12290  df-sum 12485  df-ef 12675  df-e 12676  df-sin 12677  df-cos 12678  df-pi 12680  df-dvds 12858  df-gcd 13012  df-prm 13085  df-pc 13216  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-starv 13549  df-sca 13550  df-vsca 13551  df-tset 13553  df-ple 13554  df-ds 13556  df-unif 13557  df-hom 13558  df-cco 13559  df-rest 13655  df-topn 13656  df-topgen 13672  df-pt 13673  df-prds 13676  df-xrs 13731  df-0g 13732  df-gsum 13733  df-qtop 13738  df-imas 13739  df-xps 13741  df-mre 13816  df-mrc 13817  df-acs 13819  df-mnd 14695  df-submnd 14744  df-mulg 14820  df-cntz 15121  df-cmn 15419  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-cnfld 16709  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-lp 17205  df-perf 17206  df-cn 17296  df-cnp 17297  df-haus 17384  df-cmp 17455  df-tx 17599  df-hmeo 17792  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977  df-xms 18355  df-ms 18356  df-tms 18357  df-cncf 18913  df-limc 19758  df-dv 19759  df-log 20459  df-cxp 20460  df-em 20836  df-cht 20884  df-vma 20885  df-chp 20886  df-ppi 20887  df-mu 20888
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