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Theorem selberglem1 23453
Description: Lemma for selberg 23456. Estimation of the asymptotic part of selberglem3 23455. (Contributed by Mario Carneiro, 20-May-2016.)
Hypothesis
Ref Expression
selberglem1.t  |-  T  =  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) )  /  n )
Assertion
Ref Expression
selberglem1  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1)
Distinct variable group:    x, n
Allowed substitution hints:    T( x, n)

Proof of Theorem selberglem1
StepHypRef Expression
1 fzfid 12041 . . . . . 6  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
2 elfznn 11705 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
32adantl 466 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
4 mucl 23138 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
53, 4syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
65zred 10957 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
76, 3nndivred 10575 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
87recnd 9613 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
92nnrpd 11246 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
10 rpdivcl 11233 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
119, 10sylan2 474 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
12 relogcl 22686 . . . . . . . . . 10  |-  ( ( x  /  n )  e.  RR+  ->  ( log `  ( x  /  n
) )  e.  RR )
1311, 12syl 16 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1413recnd 9613 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1514sqcld 12265 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
168, 15mulcld 9607 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  e.  CC )
171, 16fsumcl 13506 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  e.  CC )
18 2cn 10597 . . . . . . . . 9  |-  2  e.  CC
1918a1i 11 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2019, 14mulcld 9607 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( log `  (
x  /  n ) ) )  e.  CC )
2119, 20subcld 9921 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
228, 21mulcld 9607 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
231, 22fsumcl 13506 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC )
24 relogcl 22686 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2524recnd 9613 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
26 mulcl 9567 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( log `  x )  e.  CC )  -> 
( 2  x.  ( log `  x ) )  e.  CC )
2718, 25, 26sylancr 663 . . . . 5  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  e.  CC )
2817, 23, 27addsubd 9942 . . . 4  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) )  -  ( 2  x.  ( log `  x
) ) )  =  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) ) )
29 selberglem1.t . . . . . . . . 9  |-  T  =  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) )  /  n )
3029oveq2i 6288 . . . . . . . 8  |-  ( ( mmu `  n )  x.  T )  =  ( ( mmu `  n )  x.  (
( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  /  n ) )
315zcnd 10958 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  CC )
3215, 21addcld 9606 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
333nnrpd 11246 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3433rpcnne0d 11256 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  e.  CC  /\  n  =/=  0 ) )
35 divass 10216 . . . . . . . . . . 11  |-  ( ( ( mmu `  n
)  e.  CC  /\  ( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC  /\  (
n  e.  CC  /\  n  =/=  0 ) )  ->  ( ( ( mmu `  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) )  /  n
)  =  ( ( mmu `  n )  x.  ( ( ( ( log `  (
x  /  n ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  /  n
) ) )
36 div23 10217 . . . . . . . . . . 11  |-  ( ( ( mmu `  n
)  e.  CC  /\  ( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC  /\  (
n  e.  CC  /\  n  =/=  0 ) )  ->  ( ( ( mmu `  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) )  /  n
)  =  ( ( ( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
3735, 36eqtr3d 2505 . . . . . . . . . 10  |-  ( ( ( mmu `  n
)  e.  CC  /\  ( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC  /\  (
n  e.  CC  /\  n  =/=  0 ) )  ->  ( ( mmu `  n )  x.  (
( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  /  n ) )  =  ( ( ( mmu `  n )  /  n )  x.  ( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) ) )
3831, 32, 34, 37syl3anc 1223 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  x.  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) )  /  n ) )  =  ( ( ( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
398, 15, 21adddid 9611 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  ( ( ( mmu `  n )  /  n )  x.  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
4038, 39eqtrd 2503 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  x.  ( ( ( ( log `  ( x  /  n ) ) ^ 2 )  +  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) )  /  n ) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  ( ( ( mmu `  n )  /  n )  x.  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
4130, 40syl5eq 2515 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  x.  T )  =  ( ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  ( ( ( mmu `  n )  /  n )  x.  ( 2  -  (
2  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
4241sumeq2dv 13476 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  x.  T
)  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( mmu `  n )  /  n )  x.  ( ( log `  (
x  /  n ) ) ^ 2 ) )  +  ( ( ( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) ) )
431, 16, 22fsumadd 13512 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( ( mmu `  n )  /  n )  x.  ( ( log `  (
x  /  n ) ) ^ 2 ) )  +  ( ( ( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) ) )
4442, 43eqtrd 2503 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  x.  T
)  =  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  +  sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) ) )
4544oveq1d 6292 . . . 4  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) )  =  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  +  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) )  -  ( 2  x.  ( log `  x
) ) ) )
4618a1i 11 . . . . . . . 8  |-  ( x  e.  RR+  ->  2  e.  CC )
478, 14mulcld 9607 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
488, 47subcld 9921 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  -  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
491, 46, 48fsummulc2 13550 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 2  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( 2  x.  (
( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) ) ) )
501, 8, 47fsumsub 13554 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )
5150oveq2d 6293 . . . . . . 7  |-  ( x  e.  RR+  ->  ( 2  x.  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) ) )  =  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )
5249, 51eqtr3d 2505 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 2  x.  (
( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) ) )  =  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )
5319, 8mulcomd 9608 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( mmu `  n )  /  n
) )  =  ( ( ( mmu `  n )  /  n
)  x.  2 ) )
5419, 8, 14mul12d 9779 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  =  ( ( ( mmu `  n )  /  n
)  x.  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )
5553, 54oveq12d 6295 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
2  x.  ( ( mmu `  n )  /  n ) )  -  ( 2  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  2 )  -  ( ( ( mmu `  n )  /  n )  x.  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) )
5619, 8, 47subdid 10003 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( mmu `  n )  /  n )  -  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  =  ( ( 2  x.  ( ( mmu `  n )  /  n ) )  -  ( 2  x.  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )
578, 19, 20subdid 10003 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  =  ( ( ( ( mmu `  n )  /  n
)  x.  2 )  -  ( ( ( mmu `  n )  /  n )  x.  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) )
5855, 56, 573eqtr4d 2513 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( ( mmu `  n )  /  n )  -  ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  =  ( ( ( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) ) )
5958sumeq2dv 13476 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( 2  x.  (
( ( mmu `  n )  /  n
)  -  ( ( ( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) )
6052, 59eqtr3d 2505 . . . . 5  |-  ( x  e.  RR+  ->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) )
6160oveq2d 6293 . . . 4  |-  ( x  e.  RR+  ->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  +  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )  =  ( ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  + 
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) ) ) )
6228, 45, 613eqtr4d 2513 . . 3  |-  ( x  e.  RR+  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) )  =  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  +  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
6362mpteq2ia 4524 . 2  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) ) )  =  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  +  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) ) )
64 ovex 6302 . . . . 5  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  -  (
2  x.  ( log `  x ) ) )  e.  _V
6564a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  -  (
2  x.  ( log `  x ) ) )  e.  _V )
66 ovex 6302 . . . . 5  |-  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  _V
6766a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  _V )
68 mulog2sum 23445 . . . . 5  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  -  (
2  x.  ( log `  x ) ) ) )  e.  O(1)
6968a1i 11 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) ) )  e.  O(1) )
70 2ex 10598 . . . . . 6  |-  2  e.  _V
7170a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  2  e. 
_V )
72 ovex 6302 . . . . . 6  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  _V
7372a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  _V )
74 rpssre 11221 . . . . . . 7  |-  RR+  C_  RR
75 o1const 13393 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O(1) )
7674, 18, 75mp2an 672 . . . . . 6  |-  ( x  e.  RR+  |->  2 )  e.  O(1)
7776a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  2 )  e.  O(1) )
78 reex 9574 . . . . . . . . 9  |-  RR  e.  _V
7978, 74ssexi 4587 . . . . . . . 8  |-  RR+  e.  _V
8079a1i 11 . . . . . . 7  |-  ( T. 
->  RR+  e.  _V )
81 sumex 13461 . . . . . . . 8  |-  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  _V
8281a1i 11 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  _V )
83 sumex 13461 . . . . . . . 8  |-  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e. 
_V
8483a1i 11 . . . . . . 7  |-  ( ( T.  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e. 
_V )
85 eqidd 2463 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n ) )  =  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) ) )
86 eqidd 2463 . . . . . . 7  |-  ( T. 
->  ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  =  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )
8780, 82, 84, 85, 86offval2 6533 . . . . . 6  |-  ( T. 
->  ( ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
) )  oF  -  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  =  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )
88 mudivsum 23438 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O(1)
89 mulogsum 23440 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O(1)
90 o1sub 13389 . . . . . . 7  |-  ( ( ( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n ) )  e.  O(1)  /\  (
x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O(1) )  ->  (
( x  e.  RR+  |->  sum_
n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n ) )  oF  -  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O(1) )
9188, 89, 90mp2an 672 . . . . . 6  |-  ( ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  oF  -  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O(1)
9287, 91syl6eqelr 2559 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
)  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O(1) )
9371, 73, 77, 92o1mul2 13398 . . . 4  |-  ( T. 
->  ( x  e.  RR+  |->  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) )  e.  O(1) )
9465, 67, 69, 93o1add2 13397 . . 3  |-  ( T. 
->  ( x  e.  RR+  |->  ( ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  +  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) ) )  e.  O(1) )
9594trud 1383 . 2  |-  ( x  e.  RR+  |->  ( (
sum_ n  e.  (
1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  -  ( 2  x.  ( log `  x
) ) )  +  ( 2  x.  ( sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n
)  /  n )  -  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) ) ) )  e.  O(1)
9663, 95eqeltri 2546 1  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O(1)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 968    = wceq 1374   T. wtru 1375    e. wcel 1762    =/= wne 2657   _Vcvv 3108    C_ wss 3471    |-> cmpt 4500   ` cfv 5581  (class class class)co 6277    oFcof 6515   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488    - cmin 9796    / cdiv 10197   NNcn 10527   2c2 10576   ZZcz 10855   RR+crp 11211   ...cfz 11663   |_cfl 11886   ^cexp 12124   O(1)co1 13260   sum_csu 13459   logclog 22665   mmucmu 23091
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-disj 4413  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-o1 13264  df-lo1 13265  df-sum 13460  df-ef 13656  df-e 13657  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-cmp 19648  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-cxp 22668  df-em 23045  df-mu 23097
This theorem is referenced by:  selberglem2  23454
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