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Theorem selberglem1 22816
Description: Lemma for selberg 22819. Estimation of the asymptotic part of selberglem3 22818. (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 11816 . . . . . 6  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
2 elfznn 11499 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
32adantl 466 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
4 mucl 22501 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
53, 4syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
65zred 10768 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
76, 3nndivred 10391 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
87recnd 9433 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
92nnrpd 11047 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
10 rpdivcl 11034 . . . . . . . . . . 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 22049 . . . . . . . . . 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 9433 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1514sqcld 12027 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
168, 15mulcld 9427 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  e.  CC )
171, 16fsumcl 13231 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  e.  CC )
18 2cn 10413 . . . . . . . . 9  |-  2  e.  CC
1918a1i 11 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2019, 14mulcld 9427 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( log `  (
x  /  n ) ) )  e.  CC )
2119, 20subcld 9740 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
228, 21mulcld 9427 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
231, 22fsumcl 13231 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC )
24 relogcl 22049 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2524recnd 9433 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
26 mulcl 9387 . . . . . 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 9761 . . . 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 6123 . . . . . . . 8  |-  ( ( mmu `  n )  x.  T )  =  ( ( mmu `  n )  x.  (
( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  /  n ) )
315zcnd 10769 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  CC )
3215, 21addcld 9426 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
333nnrpd 11047 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3433rpcnne0d 11057 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  e.  CC  /\  n  =/=  0 ) )
35 divass 10033 . . . . . . . . . . 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 10034 . . . . . . . . . . 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 2477 . . . . . . . . . 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 1218 . . . . . . . . 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 9431 . . . . . . . . 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 2475 . . . . . . . 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 2487 . . . . . . 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 13201 . . . . . 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 13236 . . . . . 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 2475 . . . . 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 6127 . . . 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 9427 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
488, 47subcld 9740 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  -  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
491, 46, 48fsummulc2 13272 . . . . . . 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 13276 . . . . . . . 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 6128 . . . . . . 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 2477 . . . . . 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 9428 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( mmu `  n )  /  n
) )  =  ( ( ( mmu `  n )  /  n
)  x.  2 ) )
5419, 8, 14mul12d 9599 . . . . . . . . 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 6130 . . . . . . . 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 9821 . . . . . . . 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 9821 . . . . . . . 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 2485 . . . . . . 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 13201 . . . . . 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 2477 . . . . 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 6128 . . . 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 2485 . . 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 4395 . 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 6137 . . . . 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 6137 . . . . 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 22808 . . . . 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 10414 . . . . . 6  |-  2  e.  _V
7170a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  2  e. 
_V )
72 ovex 6137 . . . . . 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 11022 . . . . . . 7  |-  RR+  C_  RR
75 o1const 13118 . . . . . . 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 9394 . . . . . . . . 9  |-  RR  e.  _V
7978, 74ssexi 4458 . . . . . . . 8  |-  RR+  e.  _V
8079a1i 11 . . . . . . 7  |-  ( T. 
->  RR+  e.  _V )
81 sumex 13186 . . . . . . . 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 13186 . . . . . . . 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 2444 . . . . . . 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 2444 . . . . . . 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 6357 . . . . . 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 22801 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O(1)
89 mulogsum 22803 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O(1)
90 o1sub 13114 . . . . . . 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 2532 . . . . 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 13123 . . . 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 13122 . . 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 1378 . 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 2513 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 965    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2620   _Vcvv 2993    C_ wss 3349    e. cmpt 4371   ` cfv 5439  (class class class)co 6112    oFcof 6339   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    - cmin 9616    / cdiv 10014   NNcn 10343   2c2 10392   ZZcz 10667   RR+crp 11012   ...cfz 11458   |_cfl 11661   ^cexp 11886   O(1)co1 12985   sum_csu 13184   logclog 22028   mmucmu 22454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-pre-sup 9381  ax-addf 9382  ax-mulf 9383
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-disj 4284  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-se 4701  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-isom 5448  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-supp 6712  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-map 7237  df-pm 7238  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-fsupp 7642  df-fi 7682  df-sup 7712  df-oi 7745  df-card 8130  df-cda 8358  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-q 10975  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ioo 11325  df-ioc 11326  df-ico 11327  df-icc 11328  df-fz 11459  df-fzo 11570  df-fl 11663  df-mod 11730  df-seq 11828  df-exp 11887  df-fac 12073  df-bc 12100  df-hash 12125  df-shft 12577  df-cj 12609  df-re 12610  df-im 12611  df-sqr 12745  df-abs 12746  df-limsup 12970  df-clim 12987  df-rlim 12988  df-o1 12989  df-lo1 12990  df-sum 13185  df-ef 13374  df-e 13375  df-sin 13376  df-cos 13377  df-pi 13379  df-dvds 13557  df-gcd 13712  df-prm 13785  df-pc 13925  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-starv 14274  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-unif 14282  df-hom 14283  df-cco 14284  df-rest 14382  df-topn 14383  df-0g 14401  df-gsum 14402  df-topgen 14403  df-pt 14404  df-prds 14407  df-xrs 14461  df-qtop 14466  df-imas 14467  df-xps 14469  df-mre 14545  df-mrc 14546  df-acs 14548  df-mnd 15436  df-submnd 15486  df-mulg 15569  df-cntz 15856  df-cmn 16300  df-psmet 17831  df-xmet 17832  df-met 17833  df-bl 17834  df-mopn 17835  df-fbas 17836  df-fg 17837  df-cnfld 17841  df-top 18525  df-bases 18527  df-topon 18528  df-topsp 18529  df-cld 18645  df-ntr 18646  df-cls 18647  df-nei 18724  df-lp 18762  df-perf 18763  df-cn 18853  df-cnp 18854  df-haus 18941  df-cmp 19012  df-tx 19157  df-hmeo 19350  df-fil 19441  df-fm 19533  df-flim 19534  df-flf 19535  df-xms 19917  df-ms 19918  df-tms 19919  df-cncf 20476  df-limc 21363  df-dv 21364  df-log 22030  df-cxp 22031  df-em 22408  df-mu 22460
This theorem is referenced by:  selberglem2  22817
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