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Theorem selbergr 20665
Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
Hypothesis
Ref Expression
pntrval.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
Assertion
Ref Expression
selbergr  |-  ( x  e.  RR+  |->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) )  /  x ) )  e.  O ( 1 )
Distinct variable groups:    a, d, x    R, d, x
Allowed substitution hint:    R( a)

Proof of Theorem selbergr
StepHypRef Expression
1 reex 8782 . . . . . . 7  |-  RR  e.  _V
2 rpssre 10317 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4119 . . . . . 6  |-  RR+  e.  _V
43a1i 12 . . . . 5  |-  (  T. 
->  RR+  e.  _V )
5 ovex 5803 . . . . . 6  |-  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  e.  _V
65a1i 12 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  e.  _V )
7 ovex 5803 . . . . . 6  |-  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) )  e.  _V
87a1i 12 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) )  e.  _V )
9 eqidd 2257 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( 2  x.  ( log `  x
) ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) ) )
10 eqidd 2257 . . . . 5  |-  (  T. 
->  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) )  =  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )
114, 6, 8, 9, 10offval2 6015 . . . 4  |-  (  T. 
->  ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  -  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) ) ) )
1211trud 1320 . . 3  |-  ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  -  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) ) )
13 pntrval.r . . . . . . . . . . . 12  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
1413pntrf 20660 . . . . . . . . . . 11  |-  R : RR+
--> RR
1514ffvelrni 5584 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
1615recnd 8815 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  CC )
17 relogcl 19880 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1817recnd 8815 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1916, 18mulcld 8809 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  e.  CC )
20 fzfid 10987 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
21 elfznn 10771 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2221adantl 454 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
23 vmacl 20304 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
2422, 23syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
2524recnd 8815 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
26 rpre 10313 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  RR )
27 nndivre 9735 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  d  e.  NN )  ->  ( x  /  d
)  e.  RR )
2826, 21, 27syl2an 465 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
29 chpcl 20310 . . . . . . . . . . . 12  |-  ( ( x  /  d )  e.  RR  ->  (ψ `  ( x  /  d
) )  e.  RR )
3028, 29syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  RR )
3130recnd 8815 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  CC )
3225, 31mulcld 8809 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3320, 32fsumcl 12157 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3419, 33addcld 8808 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  e.  CC )
35 rpcn 10315 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
36 rpne0 10322 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
3734, 35, 36divcld 9490 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  e.  CC )
3824, 22nndivred 9748 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  RR )
3938recnd 8815 . . . . . . 7  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  CC )
4020, 39fsumcl 12157 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  e.  CC )
4137, 40, 18nnncan2d 9146 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( ( ( ( R `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( log `  x ) )  -  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) )  =  ( ( ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d ) ) )
42 chpcl 20310 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4326, 42syl 17 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
4443recnd 8815 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
4544, 18mulcld 8809 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
4645, 33addcld 8808 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  e.  CC )
4746, 35, 36divcld 9490 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  e.  CC )
4847, 18, 18subsub4d 9142 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( log `  x ) )  -  ( log `  x ) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( ( log `  x )  +  ( log `  x
) ) ) )
4913pntrval 20659 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
5049oveq1d 5793 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  -  x )  x.  ( log `  x
) ) )
5144, 35, 18subdird 9190 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  x.  ( log `  x ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  -  ( x  x.  ( log `  x
) ) ) )
5250, 51eqtrd 2288 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  x.  ( log `  x ) )  -  ( x  x.  ( log `  x ) ) ) )
5352oveq1d 5793 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  =  ( ( ( (ψ `  x )  x.  ( log `  x
) )  -  (
x  x.  ( log `  x ) ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) ) )
5435, 18mulcld 8809 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  x.  ( log `  x
) )  e.  CC )
5545, 33, 54addsubd 9132 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  -  ( x  x.  ( log `  x
) ) )  =  ( ( ( (ψ `  x )  x.  ( log `  x ) )  -  ( x  x.  ( log `  x
) ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) ) )
5653, 55eqtr4d 2291 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  =  ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  -  (
x  x.  ( log `  x ) ) ) )
5756oveq1d 5793 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  -  (
x  x.  ( log `  x ) ) )  /  x ) )
58 rpcnne0 10324 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
59 divsubdir 9410 . . . . . . . . . 10  |-  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  e.  CC  /\  (
x  x.  ( log `  x ) )  e.  CC  /\  ( x  e.  CC  /\  x  =/=  0 ) )  -> 
( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  -  (
x  x.  ( log `  x ) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( ( x  x.  ( log `  x ) )  /  x ) ) )
6046, 54, 58, 59syl3anc 1187 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  -  ( x  x.  ( log `  x
) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( ( x  x.  ( log `  x
) )  /  x
) ) )
6118, 35, 36divcan3d 9495 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( x  x.  ( log `  x ) )  /  x )  =  ( log `  x ) )
6261oveq2d 5794 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( ( x  x.  ( log `  x
) )  /  x
) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( log `  x ) ) )
6357, 60, 623eqtrd 2292 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( log `  x ) ) )
6463oveq1d 5793 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( log `  x ) )  =  ( ( ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( log `  x ) )  -  ( log `  x ) ) )
65182timesd 9907 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  =  ( ( log `  x
)  +  ( log `  x ) ) )
6665oveq2d 5794 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( ( log `  x
)  +  ( log `  x ) ) ) )
6748, 64, 663eqtr4d 2298 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( log `  x ) )  =  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( 2  x.  ( log `  x
) ) ) )
6867oveq1d 5793 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( ( ( ( R `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( log `  x ) )  -  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) )  =  ( ( ( ( ( (ψ `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  -  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) ) )
6935, 40mulcld 8809 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )  e.  CC )
70 divsubdir 9410 . . . . . . 7  |-  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  e.  CC  /\  ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d ) )  e.  CC  /\  (
x  e.  CC  /\  x  =/=  0 ) )  ->  ( ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  -  ( x  x. 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) ) )  /  x
)  =  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x ) ) )
7134, 69, 58, 70syl3anc 1187 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  -  (
x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) ) )  /  x
)  =  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x ) ) )
7219, 33, 69addsubassd 9131 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  -  ( x  x. 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) ) )  =  ( ( ( R `  x )  x.  ( log `  x ) )  +  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) ) ) ) )
7335adantr 453 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  CC )
7473, 39mulcld 8809 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  e.  CC )
7520, 32, 74fsumsub 12201 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  ( x  x.  (
(Λ `  d )  / 
d ) ) )  =  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( x  x.  ( (Λ `  d )  /  d
) ) ) )
7628recnd 8815 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  CC )
7725, 31, 76subdid 9189 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  (
(ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) )  =  ( ( (Λ `  d )  x.  (ψ `  ( x  /  d
) ) )  -  ( (Λ `  d )  x.  ( x  /  d
) ) ) )
7821nnrpd 10342 . . . . . . . . . . . . . . 15  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
79 rpdivcl 10329 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
8078, 79sylan2 462 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
8113pntrval 20659 . . . . . . . . . . . . . 14  |-  ( ( x  /  d )  e.  RR+  ->  ( R `
 ( x  / 
d ) )  =  ( (ψ `  (
x  /  d ) )  -  ( x  /  d ) ) )
8280, 81syl 17 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( R `  ( x  /  d
) )  =  ( (ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) )
8382oveq2d 5794 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  ( R `  ( x  /  d ) ) )  =  ( (Λ `  d )  x.  (
(ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) ) )
8422nnrpd 10342 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
85 rpcnne0 10324 . . . . . . . . . . . . . . 15  |-  ( d  e.  RR+  ->  ( d  e.  CC  /\  d  =/=  0 ) )
8684, 85syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  CC  /\  d  =/=  0 ) )
87 div12 9400 . . . . . . . . . . . . . 14  |-  ( ( x  e.  CC  /\  (Λ `  d )  e.  CC  /\  ( d  e.  CC  /\  d  =/=  0 ) )  -> 
( x  x.  (
(Λ `  d )  / 
d ) )  =  ( (Λ `  d
)  x.  ( x  /  d ) ) )
8873, 25, 86, 87syl3anc 1187 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  =  ( (Λ `  d )  x.  (
x  /  d ) ) )
8988oveq2d 5794 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (
(Λ `  d )  x.  (ψ `  ( x  /  d ) ) )  -  ( x  x.  ( (Λ `  d
)  /  d ) ) )  =  ( ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  ( (Λ `  d )  x.  ( x  /  d
) ) ) )
9077, 83, 893eqtr4d 2298 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  ( R `  ( x  /  d ) ) )  =  ( ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) )  -  ( x  x.  ( (Λ `  d
)  /  d ) ) ) )
9190sumeq2dv 12127 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  ( x  x.  (
(Λ `  d )  / 
d ) ) ) )
9220, 35, 39fsummulc2 12197 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )  =  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( x  x.  (
(Λ `  d )  / 
d ) ) )
9392oveq2d 5794 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) )  -  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) ) )  =  (
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( x  x.  ( (Λ `  d )  /  d
) ) ) )
9475, 91, 933eqtr4rd 2299 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) )  -  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) ) )  =  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  ( R `  (
x  /  d ) ) ) )
9594oveq2d 5794 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  +  ( sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  -  ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) ) ) )  =  ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  ( R `  (
x  /  d ) ) ) ) )
9672, 95eqtrd 2288 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  -  ( x  x. 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) ) )  =  ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) ) )
9796oveq1d 5793 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  -  (
x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) ) )  /  x
)  =  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) )  /  x ) )
9840, 35, 36divcan3d 9495 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )
9998oveq2d 5794 . . . . . 6  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x ) )  =  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) ) )
10071, 97, 993eqtr3rd 2297 . . . . 5  |-  ( x  e.  RR+  ->  ( ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  =  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) )  /  x ) )
10141, 68, 1003eqtr3d 2296 . . . 4  |-  ( x  e.  RR+  ->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  -  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  =  ( ( ( ( R `
 x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  ( R `  (
x  /  d ) ) ) )  /  x ) )
102101mpteq2ia 4062 . . 3  |-  ( x  e.  RR+  |->  ( ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) )  -  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( R `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  ( R `  (
x  /  d ) ) ) )  /  x ) )
10312, 102eqtri 2276 . 2  |-  ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  =  ( x  e.  RR+  |->  ( ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) )  /  x ) )
104 selberg2 20648 . . 3  |-  ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  e.  O
( 1 )
105 vmadivsum 20579 . . 3  |-  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  e.  O
( 1 )
106 o1sub 12040 . . 3  |-  ( ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x
) )  +  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  x.  (ψ `  ( x  /  d ) ) ) )  /  x
)  -  ( 2  x.  ( log `  x
) ) ) )  e.  O ( 1 )  /\  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  e.  O
( 1 ) )  ->  ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  e.  O ( 1 ) )
107104, 105, 106mp2an 656 . 2  |-  ( ( x  e.  RR+  |->  ( ( ( ( (ψ `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  /  x )  -  ( 2  x.  ( log `  x ) ) ) )  o F  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  e.  O ( 1 )
108103, 107eqeltrri 2327 1  |-  ( x  e.  RR+  |->  ( ( ( ( R `  x )  x.  ( log `  x ) )  +  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  ( R `
 ( x  / 
d ) ) ) )  /  x ) )  e.  O ( 1 )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    T. wtru 1312    = wceq 1619    e. wcel 1621    =/= wne 2419   _Vcvv 2757    e. cmpt 4037   ` cfv 4659  (class class class)co 5778    o Fcof 5996   CCcc 8689   RRcr 8690   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    - cmin 8991    / cdiv 9377   NNcn 9700   2c2 9749   RR+crp 10307   ...cfz 10734   |_cfl 10876   O ( 1 )co1 11911   sum_csu 12109   logclog 19860  Λcvma 20277  ψcchp 20278
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-disj 3954  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-of 5998  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-pm 6729  df-ixp 6772  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-fi 7119  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-7 9763  df-8 9764  df-9 9765  df-10 9766  df-n0 9919  df-z 9978  df-dec 10078  df-uz 10184  df-q 10270  df-rp 10308  df-xneg 10405  df-xadd 10406  df-xmul 10407  df-ioo 10612  df-ioc 10613  df-ico 10614  df-icc 10615  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-o1 11915  df-lo1 11916  df-sum 12110  df-ef 12297  df-e 12298  df-sin 12299  df-cos 12300  df-pi 12302  df-divides 12480  df-gcd 12634  df-prime 12707  df-pc 12838  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-starv 13171  df-sca 13172  df-vsca 13173  df-tset 13175  df-ple 13176  df-ds 13178  df-hom 13180  df-cco 13181  df-rest 13275  df-topn 13276  df-topgen 13292  df-pt 13293  df-prds 13296  df-xrs 13351  df-0g 13352  df-gsum 13353  df-qtop 13358  df-imas 13359  df-xps 13361  df-mre 13436  df-mrc 13437  df-acs 13439  df-mnd 14315  df-submnd 14364  df-mulg 14440  df-cntz 14741  df-cmn 15039  df-xmet 16321  df-met 16322  df-bl 16323  df-mopn 16324  df-cnfld 16326  df-top 16584  df-bases 16586  df-topon 16587  df-topsp 16588  df-cld 16704  df-ntr 16705  df-cls 16706  df-nei 16783  df-lp 16816  df-perf 16817  df-cn 16905  df-cnp 16906  df-haus 16991  df-cmp 17062  df-tx 17205  df-hmeo 17394  df-fbas 17468  df-fg 17469  df-fil 17489  df-fm 17581  df-flim 17582  df-flf 17583  df-xms 17833  df-ms 17834  df-tms 17835  df-cncf 18330  df-limc 19164  df-dv 19165  df-log 19862  df-cxp 19863  df-em 20235  df-cht 20282  df-vma 20283  df-chp 20284  df-ppi 20285  df-mu 20286
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