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Theorem selbergr 21250
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 9070 . . . . . . 7  |-  RR  e.  _V
2 rpssre 10611 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4340 . . . . . 6  |-  RR+  e.  _V
43a1i 11 . . . . 5  |-  (  T. 
->  RR+  e.  _V )
5 ovex 6097 . . . . . 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 11 . . . . 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 6097 . . . . . 6  |-  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) )  e.  _V
87a1i 11 . . . . 5  |-  ( (  T.  /\  x  e.  RR+ )  ->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) )  e.  _V )
9 eqidd 2436 . . . . 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 2436 . . . . 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 6313 . . . 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 1332 . . 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 21245 . . . . . . . . . . 11  |-  R : RR+
--> RR
1514ffvelrni 5860 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
1615recnd 9103 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  CC )
17 relogcl 20461 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1817recnd 9103 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1916, 18mulcld 9097 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  e.  CC )
20 fzfid 11300 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
21 elfznn 11069 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2221adantl 453 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
23 vmacl 20889 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
2524recnd 9103 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
26 rpre 10607 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  RR )
27 nndivre 10024 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  d  e.  NN )  ->  ( x  /  d
)  e.  RR )
2826, 21, 27syl2an 464 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
29 chpcl 20895 . . . . . . . . . . . 12  |-  ( ( x  /  d )  e.  RR  ->  (ψ `  ( x  /  d
) )  e.  RR )
3028, 29syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  RR )
3130recnd 9103 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  CC )
3225, 31mulcld 9097 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3320, 32fsumcl 12515 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3419, 33addcld 9096 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  e.  CC )
35 rpcn 10609 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
36 rpne0 10616 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
3734, 35, 36divcld 9779 . . . . . 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 10037 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  RR )
3938recnd 9103 . . . . . . 7  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  CC )
4020, 39fsumcl 12515 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  e.  CC )
4137, 40, 18nnncan2d 9435 . . . . 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 20895 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4326, 42syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
4443recnd 9103 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
4544, 18mulcld 9097 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
4645, 33addcld 9096 . . . . . . . . 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 9779 . . . . . . . 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 9431 . . . . . . 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 21244 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
5049oveq1d 6087 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  -  x )  x.  ( log `  x
) ) )
5144, 35, 18subdird 9479 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  x.  ( log `  x ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  -  ( x  x.  ( log `  x
) ) ) )
5250, 51eqtrd 2467 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  x.  ( log `  x ) )  -  ( x  x.  ( log `  x ) ) ) )
5352oveq1d 6087 . . . . . . . . . . 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 9097 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  x.  ( log `  x
) )  e.  CC )
5545, 33, 54addsubd 9421 . . . . . . . . . . 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 2470 . . . . . . . . . 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 6087 . . . . . . . . 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 10618 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
59 divsubdir 9699 . . . . . . . . . 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 1184 . . . . . . . . 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 9784 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( x  x.  ( log `  x ) )  /  x )  =  ( log `  x ) )
6261oveq2d 6088 . . . . . . . . 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 2471 . . . . . . . 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 6087 . . . . . . 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 10199 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  =  ( ( log `  x
)  +  ( log `  x ) ) )
6665oveq2d 6088 . . . . . . 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 2477 . . . . . 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 6087 . . . . 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 9097 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )  e.  CC )
70 divsubdir 9699 . . . . . . 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 1184 . . . . . 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 9420 . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  CC )
7473, 39mulcld 9097 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  e.  CC )
7520, 32, 74fsumsub 12559 . . . . . . . . . 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 9103 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  CC )
7725, 31, 76subdid 9478 . . . . . . . . . . . 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 10636 . . . . . . . . . . . . . . 15  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
79 rpdivcl 10623 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
8078, 79sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
8113pntrval 21244 . . . . . . . . . . . . . 14  |-  ( ( x  /  d )  e.  RR+  ->  ( R `
 ( x  / 
d ) )  =  ( (ψ `  (
x  /  d ) )  -  ( x  /  d ) ) )
8280, 81syl 16 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( R `  ( x  /  d
) )  =  ( (ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) )
8382oveq2d 6088 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  ( R `  ( x  /  d ) ) )  =  ( (Λ `  d )  x.  (
(ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) ) )
8422nnrpd 10636 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
85 rpcnne0 10618 . . . . . . . . . . . . . . 15  |-  ( d  e.  RR+  ->  ( d  e.  CC  /\  d  =/=  0 ) )
8684, 85syl 16 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( d  e.  CC  /\  d  =/=  0 ) )
87 div12 9689 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  =  ( (Λ `  d )  x.  (
x  /  d ) ) )
8988oveq2d 6088 . . . . . . . . . . . 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 2477 . . . . . . . . . . 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 12485 . . . . . . . . . 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 12555 . . . . . . . . . . 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 6088 . . . . . . . . . 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 2478 . . . . . . . . 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 6088 . . . . . . . 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 2467 . . . . . . 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 6087 . . . . . 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 9784 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )
9998oveq2d 6088 . . . . . 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 2476 . . . . 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 2475 . . . 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 4283 . . 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 2455 . 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 21233 . . 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 21164 . . 3  |-  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  e.  O
( 1 )
106 o1sub 12397 . . 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 654 . 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 2506 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 359    T. wtru 1325    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    e. cmpt 4258   ` cfv 5445  (class class class)co 6072    o Fcof 6294   CCcc 8977   RRcr 8978   0cc0 8979   1c1 8980    + caddc 8982    x. cmul 8984    - cmin 9280    / cdiv 9666   NNcn 9989   2c2 10038   RR+crp 10601   ...cfz 11032   |_cfl 11189   O ( 1 )co1 12268   sum_csu 12467   logclog 20440  Λcvma 20862  ψcchp 20863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-o1 12272  df-lo1 12273  df-sum 12468  df-ef 12658  df-e 12659  df-sin 12660  df-cos 12661  df-pi 12663  df-dvds 12841  df-gcd 12995  df-prm 13068  df-pc 13199  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-cxp 20443  df-em 20819  df-cht 20867  df-vma 20868  df-chp 20869  df-ppi 20870  df-mu 20871
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