MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  selbergr Unicode version

Theorem selbergr 20644
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 8761 . . . . . . 7  |-  RR  e.  _V
2 rpssre 10296 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4099 . . . . . 6  |-  RR+  e.  _V
43a1i 12 . . . . 5  |-  (  T. 
->  RR+  e.  _V )
5 ovex 5782 . . . . . 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 5782 . . . . . 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 5994 . . . 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 20639 . . . . . . . . . . 11  |-  R : RR+
--> RR
1514ffvelrni 5563 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
1615recnd 8794 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  CC )
17 relogcl 19859 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1817recnd 8794 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1916, 18mulcld 8788 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  e.  CC )
20 fzfid 10966 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
21 elfznn 10750 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2221adantl 454 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
23 vmacl 20283 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
2422, 23syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
2524recnd 8794 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
26 rpre 10292 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  RR )
27 nndivre 9714 . . . . . . . . . . . . 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 20289 . . . . . . . . . . . 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 8794 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  CC )
3225, 31mulcld 8788 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3320, 32fsumcl 12136 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3419, 33addcld 8787 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  e.  CC )
35 rpcn 10294 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
36 rpne0 10301 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
3734, 35, 36divcld 9469 . . . . . 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 9727 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  RR )
3938recnd 8794 . . . . . . 7  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  CC )
4020, 39fsumcl 12136 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  e.  CC )
4137, 40, 18nnncan2d 9125 . . . . 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 20289 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4326, 42syl 17 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
4443recnd 8794 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
4544, 18mulcld 8788 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
4645, 33addcld 8787 . . . . . . . . 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 9469 . . . . . . . 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 9121 . . . . . . 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 20638 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
5049oveq1d 5772 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  -  x )  x.  ( log `  x
) ) )
5144, 35, 18subdird 9169 . . . . . . . . . . . . 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 5772 . . . . . . . . . . 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 8788 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  x.  ( log `  x
) )  e.  CC )
5545, 33, 54addsubd 9111 . . . . . . . . . . 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 5772 . . . . . . . . 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 10303 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
59 divsubdir 9389 . . . . . . . . . 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 9474 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( x  x.  ( log `  x ) )  /  x )  =  ( log `  x ) )
6261oveq2d 5773 . . . . . . . . 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 5772 . . . . . . 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 9886 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  =  ( ( log `  x
)  +  ( log `  x ) ) )
6665oveq2d 5773 . . . . . . 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 5772 . . . . 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 8788 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )  e.  CC )
70 divsubdir 9389 . . . . . . 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 9110 . . . . . . . 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 8788 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  e.  CC )
7520, 32, 74fsumsub 12180 . . . . . . . . . 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 8794 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  CC )
7725, 31, 76subdid 9168 . . . . . . . . . . . 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 10321 . . . . . . . . . . . . . . 15  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
79 rpdivcl 10308 . . . . . . . . . . . . . . 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 20638 . . . . . . . . . . . . . 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 5773 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  ( R `  ( x  /  d ) ) )  =  ( (Λ `  d )  x.  (
(ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) ) )
8422nnrpd 10321 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
85 rpcnne0 10303 . . . . . . . . . . . . . . 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 9379 . . . . . . . . . . . . . 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 5773 . . . . . . . . . . . 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 12106 . . . . . . . . . 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 12176 . . . . . . . . . . 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 5773 . . . . . . . . . 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 5773 . . . . . . . 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 5772 . . . . . 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 9474 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )
9998oveq2d 5773 . . . . . 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 4042 . . 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 20627 . . 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 20558 . . 3  |-  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  e.  O
( 1 )
106 o1sub 12019 . . 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 2740    e. cmpt 4017   ` cfv 4638  (class class class)co 5757    o Fcof 5975   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   RR+crp 10286   ...cfz 10713   |_cfl 10855   O ( 1 )co1 11890   sum_csu 12088   logclog 19839  Λcvma 20256  ψcchp 20257
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 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-disj 3935  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-o1 11894  df-lo1 11895  df-sum 12089  df-ef 12276  df-e 12277  df-sin 12278  df-cos 12279  df-pi 12281  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-cmp 17041  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-cxp 19842  df-em 20214  df-cht 20261  df-vma 20262  df-chp 20263  df-ppi 20264  df-mu 20265
  Copyright terms: Public domain W3C validator