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Theorem selbergr 22817
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 9373 . . . . . . 7  |-  RR  e.  _V
2 rpssre 11001 . . . . . . 7  |-  RR+  C_  RR
31, 2ssexi 4437 . . . . . 6  |-  RR+  e.  _V
43a1i 11 . . . . 5  |-  ( T. 
->  RR+  e.  _V )
5 ovex 6116 . . . . . 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 6116 . . . . . 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 2444 . . . . 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 2444 . . . . 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 6336 . . . 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 ) ) ) )  oF  -  ( 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 1378 . . 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 ) ) ) )  oF  -  ( 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 22812 . . . . . . . . . . 11  |-  R : RR+
--> RR
1514ffvelrni 5842 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  RR )
1615recnd 9412 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( R `
 x )  e.  CC )
17 relogcl 22027 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( log `  x )  e.  RR )
1817recnd 9412 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( log `  x )  e.  CC )
1916, 18mulcld 9406 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  e.  CC )
20 fzfid 11795 . . . . . . . . 9  |-  ( x  e.  RR+  ->  ( 1 ... ( |_ `  x ) )  e. 
Fin )
21 elfznn 11478 . . . . . . . . . . . . 13  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  NN )
2221adantl 466 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  NN )
23 vmacl 22456 . . . . . . . . . . . 12  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
2422, 23syl 16 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  RR )
2524recnd 9412 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (Λ `  d
)  e.  CC )
26 rpre 10997 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  x  e.  RR )
27 nndivre 10357 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  d  e.  NN )  ->  ( x  /  d
)  e.  RR )
2826, 21, 27syl2an 477 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR )
29 chpcl 22462 . . . . . . . . . . . 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 9412 . . . . . . . . . 10  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  (ψ `  (
x  /  d ) )  e.  CC )
3225, 31mulcld 9406 . . . . . . . . 9  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3320, 32fsumcl 13210 . . . . . . . 8  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) )  e.  CC )
3419, 33addcld 9405 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( ( R `  x
)  x.  ( log `  x ) )  + 
sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  x.  (ψ `  ( x  /  d
) ) ) )  e.  CC )
35 rpcn 10999 . . . . . . 7  |-  ( x  e.  RR+  ->  x  e.  CC )
36 rpne0 11006 . . . . . . 7  |-  ( x  e.  RR+  ->  x  =/=  0 )
3734, 35, 36divcld 10107 . . . . . 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 10370 . . . . . . . 8  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  RR )
3938recnd 9412 . . . . . . 7  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  /  d
)  e.  CC )
4020, 39fsumcl 13210 . . . . . 6  |-  ( x  e.  RR+  ->  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  e.  CC )
4137, 40, 18nnncan2d 9754 . . . . 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 22462 . . . . . . . . . . . . 13  |-  ( x  e.  RR  ->  (ψ `  x )  e.  RR )
4326, 42syl 16 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  RR )
4443recnd 9412 . . . . . . . . . . 11  |-  ( x  e.  RR+  ->  (ψ `  x )  e.  CC )
4544, 18mulcld 9406 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( (ψ `  x )  x.  ( log `  x ) )  e.  CC )
4645, 33addcld 9405 . . . . . . . . 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 10107 . . . . . . . 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 9750 . . . . . . 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 22811 . . . . . . . . . . . . . 14  |-  ( x  e.  RR+  ->  ( R `
 x )  =  ( (ψ `  x
)  -  x ) )
5049oveq1d 6106 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  -  x )  x.  ( log `  x
) ) )
5144, 35, 18subdird 9801 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  ->  ( ( (ψ `  x )  -  x )  x.  ( log `  x ) )  =  ( ( (ψ `  x )  x.  ( log `  x ) )  -  ( x  x.  ( log `  x
) ) ) )
5250, 51eqtrd 2475 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( ( R `  x )  x.  ( log `  x
) )  =  ( ( (ψ `  x
)  x.  ( log `  x ) )  -  ( x  x.  ( log `  x ) ) ) )
5352oveq1d 6106 . . . . . . . . . . 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 9406 . . . . . . . . . . . 12  |-  ( x  e.  RR+  ->  ( x  x.  ( log `  x
) )  e.  CC )
5545, 33, 54addsubd 9740 . . . . . . . . . . 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 2478 . . . . . . . . . 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 6106 . . . . . . . . 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 11008 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( x  e.  CC  /\  x  =/=  0 ) )
59 divsubdir 10027 . . . . . . . . . 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 1218 . . . . . . . . 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 10112 . . . . . . . . . 10  |-  ( x  e.  RR+  ->  ( ( x  x.  ( log `  x ) )  /  x )  =  ( log `  x ) )
6261oveq2d 6107 . . . . . . . . 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 2479 . . . . . . . 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 6106 . . . . . . 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 10567 . . . . . . . 8  |-  ( x  e.  RR+  ->  ( 2  x.  ( log `  x
) )  =  ( ( log `  x
)  +  ( log `  x ) ) )
6665oveq2d 6107 . . . . . . 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 2485 . . . . . 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 6106 . . . . 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 9406 . . . . . . 7  |-  ( x  e.  RR+  ->  ( x  x.  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )  e.  CC )
70 divsubdir 10027 . . . . . . 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 1218 . . . . . 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 9739 . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  x  e.  CC )
7473, 39mulcld 9406 . . . . . . . . . . 11  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  e.  CC )
7520, 32, 74fsumsub 13255 . . . . . . . . . 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 9412 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  CC )
7725, 31, 76subdid 9800 . . . . . . . . . . . 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 11026 . . . . . . . . . . . . . . 15  |-  ( d  e.  ( 1 ... ( |_ `  x
) )  ->  d  e.  RR+ )
79 rpdivcl 11013 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  RR+ )  ->  (
x  /  d )  e.  RR+ )
8078, 79sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  /  d )  e.  RR+ )
8113pntrval 22811 . . . . . . . . . . . . . 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 6107 . . . . . . . . . . . 12  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( (Λ `  d )  x.  ( R `  ( x  /  d ) ) )  =  ( (Λ `  d )  x.  (
(ψ `  ( x  /  d ) )  -  ( x  / 
d ) ) ) )
8422nnrpd 11026 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  d  e.  RR+ )
85 rpcnne0 11008 . . . . . . . . . . . . . . 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 10016 . . . . . . . . . . . . . 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 1218 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR+  /\  d  e.  ( 1 ... ( |_ `  x ) ) )  ->  ( x  x.  ( (Λ `  d
)  /  d ) )  =  ( (Λ `  d )  x.  (
x  /  d ) ) )
8988oveq2d 6107 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 13180 . . . . . . . . . 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 13251 . . . . . . . . . . 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 6107 . . . . . . . . . 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 2486 . . . . . . . . 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 6107 . . . . . . . 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 2475 . . . . . . 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 6106 . . . . . 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 10112 . . . . . . 7  |-  ( x  e.  RR+  ->  ( ( x  x.  sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d ) )  /  x )  =  sum_ d  e.  ( 1 ... ( |_
`  x ) ) ( (Λ `  d
)  /  d ) )
9998oveq2d 6107 . . . . . 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 2484 . . . . 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 2483 . . . 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 4374 . . 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 2463 . 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 ) ) ) )  oF  -  ( 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 22800 . . 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 22731 . . 3  |-  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d )  /  d )  -  ( log `  x ) ) )  e.  O(1)
106 o1sub 13093 . . 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 ) ) ) )  oF  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  e.  O(1) )
107104, 105, 106mp2an 672 . 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 ) ) ) )  oF  -  ( x  e.  RR+  |->  ( sum_ d  e.  ( 1 ... ( |_ `  x ) ) ( (Λ `  d
)  /  d )  -  ( log `  x
) ) ) )  e.  O(1)
108103, 107eqeltrri 2514 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1369   T. wtru 1370    e. wcel 1756    =/= wne 2606   _Vcvv 2972    e. cmpt 4350   ` cfv 5418  (class class class)co 6091    oFcof 6318   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595    / cdiv 9993   NNcn 10322   2c2 10371   RR+crp 10991   ...cfz 11437   |_cfl 11640   O(1)co1 12964   sum_csu 13163   logclog 22006  Λcvma 22429  ψcchp 22430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-disj 4263  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-ioc 11305  df-ico 11306  df-icc 11307  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-fac 12052  df-bc 12079  df-hash 12104  df-shft 12556  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-limsup 12949  df-clim 12966  df-rlim 12967  df-o1 12968  df-lo1 12969  df-sum 13164  df-ef 13353  df-e 13354  df-sin 13355  df-cos 13356  df-pi 13358  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-mulg 15548  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-fbas 17814  df-fg 17815  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740  df-perf 18741  df-cn 18831  df-cnp 18832  df-haus 18919  df-cmp 18990  df-tx 19135  df-hmeo 19328  df-fil 19419  df-fm 19511  df-flim 19512  df-flf 19513  df-xms 19895  df-ms 19896  df-tms 19897  df-cncf 20454  df-limc 21341  df-dv 21342  df-log 22008  df-cxp 22009  df-em 22386  df-cht 22434  df-vma 22435  df-chp 22436  df-ppi 22437  df-mu 22438
This theorem is referenced by: (None)
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