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Theorem selberglem1 22679
Description: Lemma for selberg 22682. Estimation of the asymptotic part of selberglem3 22681. (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 11779 . . . . . 6  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
2 elfznn 11465 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
32adantl 463 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
4 mucl 22364 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
53, 4syl 16 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
65zred 10735 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
76, 3nndivred 10358 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
87recnd 9400 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
92nnrpd 11014 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
10 rpdivcl 11001 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
119, 10sylan2 471 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
12 relogcl 21912 . . . . . . . . . 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 9400 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1514sqcld 11990 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
168, 15mulcld 9394 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  e.  CC )
171, 16fsumcl 13194 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  e.  CC )
18 2cn 10380 . . . . . . . . 9  |-  2  e.  CC
1918a1i 11 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2019, 14mulcld 9394 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( log `  (
x  /  n ) ) )  e.  CC )
2119, 20subcld 9707 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
228, 21mulcld 9394 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
231, 22fsumcl 13194 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC )
24 relogcl 21912 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2524recnd 9400 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
26 mulcl 9354 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( log `  x )  e.  CC )  -> 
( 2  x.  ( log `  x ) )  e.  CC )
2718, 25, 26sylancr 656 . . . . 5  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  e.  CC )
2817, 23, 27addsubd 9728 . . . 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 6091 . . . . . . . 8  |-  ( ( mmu `  n )  x.  T )  =  ( ( mmu `  n )  x.  (
( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  /  n ) )
315zcnd 10736 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  CC )
3215, 21addcld 9393 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
333nnrpd 11014 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3433rpcnne0d 11024 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  e.  CC  /\  n  =/=  0 ) )
35 divass 10000 . . . . . . . . . . 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 10001 . . . . . . . . . . 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 2467 . . . . . . . . . 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 1211 . . . . . . . . 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 9398 . . . . . . . . 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 2465 . . . . . . . 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 2477 . . . . . . 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 13164 . . . . . 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 13199 . . . . . 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 2465 . . . . 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 6095 . . . 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 9394 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
488, 47subcld 9707 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  -  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
491, 46, 48fsummulc2 13234 . . . . . . 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 13238 . . . . . . . 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 6096 . . . . . . 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 2467 . . . . . 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 9395 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( mmu `  n )  /  n
) )  =  ( ( ( mmu `  n )  /  n
)  x.  2 ) )
5419, 8, 14mul12d 9566 . . . . . . . . 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 6098 . . . . . . . 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 9788 . . . . . . . 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 9788 . . . . . . . 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 2475 . . . . . . 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 13164 . . . . . 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 2467 . . . . 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 6096 . . . 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 2475 . . 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 4362 . 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 6105 . . . . 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 6105 . . . . 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 22671 . . . . 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 10381 . . . . . 6  |-  2  e.  _V
7170a1i 11 . . . . 5  |-  ( ( T.  /\  x  e.  RR+ )  ->  2  e. 
_V )
72 ovex 6105 . . . . . 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 10989 . . . . . . 7  |-  RR+  C_  RR
75 o1const 13081 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O(1) )
7674, 18, 75mp2an 665 . . . . . 6  |-  ( x  e.  RR+  |->  2 )  e.  O(1)
7776a1i 11 . . . . 5  |-  ( T. 
->  ( x  e.  RR+  |->  2 )  e.  O(1) )
78 reex 9361 . . . . . . . . 9  |-  RR  e.  _V
7978, 74ssexi 4425 . . . . . . . 8  |-  RR+  e.  _V
8079a1i 11 . . . . . . 7  |-  ( T. 
->  RR+  e.  _V )
81 sumex 13149 . . . . . . . 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 13149 . . . . . . . 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 2434 . . . . . . 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 2434 . . . . . . 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 6325 . . . . . 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 22664 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O(1)
89 mulogsum 22666 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O(1)
90 o1sub 13077 . . . . . . 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 665 . . . . . 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 2522 . . . . 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 13086 . . . 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 13085 . . 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 1371 . 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 2503 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 958    = wceq 1362   T. wtru 1363    e. wcel 1755    =/= wne 2596   _Vcvv 2962    C_ wss 3316    e. cmpt 4338   ` cfv 5406  (class class class)co 6080    oFcof 6307   CCcc 9268   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   ZZcz 10634   RR+crp 10979   ...cfz 11424   |_cfl 11624   ^cexp 11849   O(1)co1 12948   sum_csu 13147   logclog 21891   mmucmu 22317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-disj 4251  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-o1 12952  df-lo1 12953  df-sum 13148  df-ef 13336  df-e 13337  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894  df-em 22271  df-mu 22323
This theorem is referenced by:  selberglem2  22680
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