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Theorem selberglem1 20710
Description: Lemma for selberg 20713. Estimation of the asymptotic part of selberglem3 20712. (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 11051 . . . . . 6  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
2 elfznn 10835 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  NN )
32adantl 452 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  NN )
4 mucl 20395 . . . . . . . . . . 11  |-  ( n  e.  NN  ->  (
mmu `  n )  e.  ZZ )
53, 4syl 15 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  ZZ )
65zred 10133 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  RR )
76, 3nndivred 9810 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  RR )
87recnd 8877 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
mmu `  n )  /  n )  e.  CC )
92nnrpd 10405 . . . . . . . . . . 11  |-  ( n  e.  ( 1 ... ( |_ `  x
) )  ->  n  e.  RR+ )
10 rpdivcl 10392 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  RR+ )  ->  (
x  /  n )  e.  RR+ )
119, 10sylan2 460 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  n )  e.  RR+ )
12 relogcl 19948 . . . . . . . . . 10  |-  ( ( x  /  n )  e.  RR+  ->  ( log `  ( x  /  n
) )  e.  RR )
1311, 12syl 15 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  RR )
1413recnd 8877 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( log `  ( x  /  n
) )  e.  CC )
1514sqcld 11259 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( ( log `  ( x  /  n ) ) ^
2 )  e.  CC )
168, 15mulcld 8871 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  e.  CC )
171, 16fsumcl 12222 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( ( log `  ( x  /  n ) ) ^ 2 ) )  e.  CC )
18 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
1918a1i 10 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  2  e.  CC )
2019, 14mulcld 8871 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( log `  (
x  /  n ) ) )  e.  CC )
2119, 20subcld 9173 . . . . . . 7  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
228, 21mulcld 8871 . . . . . 6  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
231, 22fsumcl 12222 . . . . 5  |-  ( x  e.  RR+  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  e.  CC )
24 relogcl 19948 . . . . . . 7  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
2524recnd 8877 . . . . . 6  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
26 mulcl 8837 . . . . . 6  |-  ( ( 2  e.  CC  /\  ( log `  x )  e.  CC )  -> 
( 2  x.  ( log `  x ) )  e.  CC )
2718, 25, 26sylancr 644 . . . . 5  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  e.  CC )
2817, 23, 27addsubd 9194 . . . 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 5885 . . . . . . . 8  |-  ( ( mmu `  n )  x.  T )  =  ( ( mmu `  n )  x.  (
( ( ( log `  ( x  /  n
) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  (
x  /  n ) ) ) ) )  /  n ) )
315zcnd 10134 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( mmu `  n )  e.  CC )
3215, 21addcld 8870 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( log `  (
x  /  n ) ) ^ 2 )  +  ( 2  -  ( 2  x.  ( log `  ( x  /  n ) ) ) ) )  e.  CC )
333nnrpd 10405 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  n  e.  RR+ )
3433rpcnne0d 10415 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( n  e.  CC  /\  n  =/=  0 ) )
35 divass 9458 . . . . . . . . . . 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 9459 . . . . . . . . . . 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 2330 . . . . . . . . . 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 1182 . . . . . . . . 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 8875 . . . . . . . . 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 2328 . . . . . . . 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 2340 . . . . . . 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 12192 . . . . . 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 12227 . . . . . 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 2328 . . . . 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 5889 . . . 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 10 . . . . . . . 8  |-  ( x  e.  RR+  ->  2  e.  CC )
478, 14mulcld 8871 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  x.  ( log `  (
x  /  n ) ) )  e.  CC )
488, 47subcld 9173 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
( mmu `  n
)  /  n )  -  ( ( ( mmu `  n )  /  n )  x.  ( log `  (
x  /  n ) ) ) )  e.  CC )
491, 46, 48fsummulc2 12262 . . . . . . 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 12266 . . . . . . . 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 5890 . . . . . . 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 2330 . . . . . 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 8872 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  n  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( 2  x.  ( ( mmu `  n )  /  n
) )  =  ( ( ( mmu `  n )  /  n
)  x.  2 ) )
5419, 8, 14mul12d 9037 . . . . . . . . 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 5892 . . . . . . . 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 9251 . . . . . . . 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 9251 . . . . . . . 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 2338 . . . . . . 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 12192 . . . . . 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 2330 . . . . 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 5890 . . . 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 2338 . . 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 4118 . 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 5899 . . . . 5  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( ( mmu `  n
)  /  n )  x.  ( ( log `  ( x  /  n
) ) ^ 2 ) )  -  (
2  x.  ( log `  x ) ) )  e.  _V
6564a1i 10 . . . 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 5899 . . . . 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 10 . . . 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 20702 . . . . 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 10 . . . 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 ) )
7018elexi 2810 . . . . . 6  |-  2  e.  _V
7170a1i 10 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  2  e. 
_V )
72 ovex 5899 . . . . . 6  |-  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  /  n )  -  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  _V
7372a1i 10 . . . . 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 10380 . . . . . . 7  |-  RR+  C_  RR
75 o1const 12109 . . . . . . 7  |-  ( (
RR+  C_  RR  /\  2  e.  CC )  ->  (
x  e.  RR+  |->  2 )  e.  O ( 1 ) )
7674, 18, 75mp2an 653 . . . . . 6  |-  ( x  e.  RR+  |->  2 )  e.  O ( 1 )
7776a1i 10 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  2 )  e.  O
( 1 ) )
78 reex 8844 . . . . . . . . 9  |-  RR  e.  _V
7978, 74ssexi 4175 . . . . . . . 8  |-  RR+  e.  _V
8079a1i 10 . . . . . . 7  |-  (  T. 
->  RR+  e.  _V )
81 sumex 12176 . . . . . . . 8  |-  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  _V
8281a1i 10 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
)  e.  _V )
83 sumex 12176 . . . . . . . 8  |-  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e. 
_V
8483a1i 10 . . . . . . 7  |-  ( (  T.  /\  x  e.  RR+ )  ->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) )  e. 
_V )
85 eqidd 2297 . . . . . . 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 2297 . . . . . . 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 6111 . . . . . 6  |-  (  T. 
->  ( ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( mmu `  n )  /  n
) )  o F  -  ( 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 20695 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  e.  O
( 1 )
89 mulogsum 20697 . . . . . . 7  |-  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) )  e.  O ( 1 )
90 o1sub 12105 . . . . . . 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 ) )  o F  -  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O ( 1 ) )
9188, 89, 90mp2an 653 . . . . . 6  |-  ( ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_ `  x ) ) ( ( mmu `  n )  /  n
) )  o F  -  ( x  e.  RR+  |->  sum_ n  e.  ( 1 ... ( |_
`  x ) ) ( ( ( mmu `  n )  /  n
)  x.  ( log `  ( x  /  n
) ) ) ) )  e.  O ( 1 )
9287, 91syl6eqelr 2385 . . . . 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 12114 . . . 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 12113 . . 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 1314 . 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 2366 1  |-  ( x  e.  RR+  |->  ( sum_ n  e.  ( 1 ... ( |_ `  x
) ) ( ( mmu `  n )  x.  T )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O
( 1 )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   ZZcz 10040   RR+crp 10370   ...cfz 10798   |_cfl 10940   ^cexp 11120   O ( 1 )co1 11976   sum_csu 12174   logclog 19928   mmucmu 20348
This theorem is referenced by:  selberglem2  20711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-o1 11980  df-lo1 11981  df-sum 12175  df-ef 12365  df-e 12366  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-em 20303  df-mu 20354
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