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Theorem pntsval2 22712
Description: The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.)
Hypothesis
Ref Expression
pntsval.1  |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) ) )
Assertion
Ref Expression
pntsval2  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) ) )
Distinct variable groups:    i, a, m, n, y, A    S, m, n, y
Allowed substitution hints:    S( i, a)

Proof of Theorem pntsval2
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 pntsval.1 . . 3  |-  S  =  ( a  e.  RR  |->  sum_ i  e.  ( 1 ... ( |_ `  a ) ) ( (Λ `  i )  x.  ( ( log `  i
)  +  (ψ `  ( a  /  i
) ) ) ) )
21pntsval 22708 . 2  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
3 elfznn 11467 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 463 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 vmacl 22343 . . . . . 6  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
64, 5syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
76recnd 9402 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
84nnrpd 11016 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  RR+ )
98relogcld 21959 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  RR )
109recnd 9402 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  CC )
11 simpl 454 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
1211, 4nndivred 10360 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  n )  e.  RR )
13 chpcl 22349 . . . . . 6  |-  ( ( A  /  n )  e.  RR  ->  (ψ `  ( A  /  n
) )  e.  RR )
1412, 13syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  RR )
1514recnd 9402 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  CC )
167, 10, 15adddid 9400 . . 3  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) )  =  ( ( (Λ `  n
)  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) ) ) )
1716sumeq2dv 13166 . 2  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) )  =  sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( ( (Λ `  n )  x.  ( log `  n
) )  +  ( (Λ `  n )  x.  (ψ `  ( A  /  n ) ) ) ) )
18 fveq2 5681 . . . . . . 7  |-  ( n  =  m  ->  (Λ `  n )  =  (Λ `  m ) )
19 oveq2 6090 . . . . . . . 8  |-  ( n  =  m  ->  ( A  /  n )  =  ( A  /  m
) )
2019fveq2d 5685 . . . . . . 7  |-  ( n  =  m  ->  (ψ `  ( A  /  n
) )  =  (ψ `  ( A  /  m
) ) )
2118, 20oveq12d 6100 . . . . . 6  |-  ( n  =  m  ->  (
(Λ `  n )  x.  (ψ `  ( A  /  n ) ) )  =  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) ) )
2221cbvsumv 13159 . . . . 5  |-  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  = 
sum_ m  e.  (
1 ... ( |_ `  A ) ) ( (Λ `  m )  x.  (ψ `  ( A  /  m ) ) )
23 fzfid 11781 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  m ) ) )  e.  Fin )
24 elfznn 11467 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
2524adantl 463 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
26 vmacl 22343 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
2725, 26syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  RR )
2827recnd 9402 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  CC )
29 elfznn 11467 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) )  ->  k  e.  NN )
3029adantl 463 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  NN )
31 vmacl 22343 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
3230, 31syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  RR )
3332recnd 9402 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  CC )
3423, 28, 33fsummulc2 13236 . . . . . . . 8  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  m
)  x.  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) ) (Λ `  k )
)  =  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) ) ( (Λ `  m
)  x.  (Λ `  k
) ) )
35 simpl 454 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
3635, 25nndivred 10360 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR )
37 chpval 22347 . . . . . . . . . 10  |-  ( ( A  /  m )  e.  RR  ->  (ψ `  ( A  /  m
) )  =  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m
) ) ) (Λ `  k ) )
3836, 37syl 16 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  m ) )  =  sum_ k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) (Λ `  k )
)
3938oveq2d 6098 . . . . . . . 8  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) )  =  ( (Λ `  m
)  x.  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) ) (Λ `  k )
) )
4030nncnd 10328 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  CC )
4124ad2antlr 721 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  NN )
4241nncnd 10328 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  CC )
4341nnne0d 10356 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  =/=  0 )
4440, 42, 43divcan3d 10102 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  ( (
m  x.  k )  /  m )  =  k )
4544fveq2d 5685 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  (
( m  x.  k
)  /  m ) )  =  (Λ `  k
) )
4645oveq2d 6098 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  ( (Λ `  m )  x.  (Λ `  ( ( m  x.  k )  /  m
) ) )  =  ( (Λ `  m
)  x.  (Λ `  k
) ) )
4746sumeq2dv 13166 . . . . . . . 8  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) )  =  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m
) ) ) ( (Λ `  m )  x.  (Λ `  k )
) )
4834, 39, 473eqtr4d 2477 . . . . . . 7  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) )  = 
sum_ k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
4948sumeq2dv 13166 . . . . . 6  |-  ( A  e.  RR  ->  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) )  = 
sum_ m  e.  (
1 ... ( |_ `  A ) ) sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m
) ) ) ( (Λ `  m )  x.  (Λ `  ( (
m  x.  k )  /  m ) ) ) )
50 oveq1 6089 . . . . . . . . 9  |-  ( n  =  ( m  x.  k )  ->  (
n  /  m )  =  ( ( m  x.  k )  /  m ) )
5150fveq2d 5685 . . . . . . . 8  |-  ( n  =  ( m  x.  k )  ->  (Λ `  ( n  /  m
) )  =  (Λ `  ( ( m  x.  k )  /  m
) ) )
5251oveq2d 6098 . . . . . . 7  |-  ( n  =  ( m  x.  k )  ->  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  =  ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
53 id 22 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR )
54 ssrab2 3427 . . . . . . . . . . . 12  |-  { y  e.  NN  |  y 
||  n }  C_  NN
55 simpr 458 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  { y  e.  NN  | 
y  ||  n }
)
5654, 55sseldi 3344 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  NN )
5756, 26syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  m
)  e.  RR )
58 dvdsdivcl 22408 . . . . . . . . . . . . 13  |-  ( ( n  e.  NN  /\  m  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  m )  e.  { y  e.  NN  |  y  ||  n } )
594, 58sylan 468 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  m )  e.  {
y  e.  NN  | 
y  ||  n }
)
6054, 59sseldi 3344 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  m )  e.  NN )
61 vmacl 22343 . . . . . . . . . . 11  |-  ( ( n  /  m )  e.  NN  ->  (Λ `  ( n  /  m
) )  e.  RR )
6260, 61syl 16 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  (
n  /  m ) )  e.  RR )
6357, 62remulcld 9404 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  RR )
6463recnd 9402 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  CC )
6564anasss 642 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( n  e.  (
1 ... ( |_ `  A ) )  /\  m  e.  { y  e.  NN  |  y  ||  n } ) )  -> 
( (Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  e.  CC )
6652, 53, 65dvdsflsumcom 22415 . . . . . 6  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) )
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  =  sum_ m  e.  ( 1 ... ( |_
`  A ) )
sum_ k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
6749, 66eqtr4d 2470 . . . . 5  |-  ( A  e.  RR  ->  sum_ m  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ m  e.  { y  e.  NN  |  y  ||  n }  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) ) )
6822, 67syl5eq 2479 . . . 4  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  = 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ m  e.  { y  e.  NN  |  y  ||  n }  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) ) )
6968oveq2d 6098 . . 3  |-  ( A  e.  RR  ->  ( sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n )  x.  ( log `  n
) )  +  sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( (Λ `  n )  x.  (ψ `  ( A  /  n
) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ n  e.  (
1 ... ( |_ `  A ) ) sum_ m  e.  { y  e.  NN  |  y  ||  n }  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) ) ) )
70 fzfid 11781 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
717, 10mulcld 9396 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  ( log `  n ) )  e.  CC )
727, 15mulcld 9396 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  e.  CC )
7370, 71, 72fsumadd 13201 . . 3  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ n  e.  (
1 ... ( |_ `  A ) ) ( (Λ `  n )  x.  (ψ `  ( A  /  n ) ) ) ) )
74 fzfid 11781 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e.  Fin )
75 sgmss 22331 . . . . . . . 8  |-  ( n  e.  NN  ->  { y  e.  NN  |  y 
||  n }  C_  ( 1 ... n
) )
764, 75syl 16 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  C_  ( 1 ... n ) )
77 ssfi 7523 . . . . . . 7  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
7874, 76, 77syl2anc 656 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
7978, 63fsumrecl 13197 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ m  e.  {
y  e.  NN  | 
y  ||  n } 
( (Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  e.  RR )
8079recnd 9402 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  sum_ m  e.  {
y  e.  NN  | 
y  ||  n } 
( (Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  e.  CC )
8170, 71, 80fsumadd 13201 . . 3  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) )  =  ( sum_ n  e.  ( 1 ... ( |_ `  A
) ) ( (Λ `  n )  x.  ( log `  n ) )  +  sum_ n  e.  ( 1 ... ( |_
`  A ) )
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) ) )
8269, 73, 813eqtr4d 2477 . 2  |-  ( A  e.  RR  ->  sum_ n  e.  ( 1 ... ( |_ `  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  +  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) ) )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) ) )
832, 17, 823eqtrd 2471 1  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( ( (Λ `  n
)  x.  ( log `  n ) )  + 
sum_ m  e.  { y  e.  NN  |  y 
||  n }  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1757   {crab 2711    C_ wss 3318   class class class wbr 4282    e. cmpt 4340   ` cfv 5408  (class class class)co 6082   Fincfn 7300   CCcc 9270   RRcr 9271   1c1 9273    + caddc 9275    x. cmul 9277    / cdiv 9983   NNcn 10312   ...cfz 11426   |_cfl 11626   sum_csu 13149    || cdivides 13520   logclog 21893  Λcvma 22316  ψcchp 22317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-rep 4393  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-inf2 7837  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349  ax-pre-sup 9350  ax-addf 9351  ax-mulf 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-iin 4164  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-se 4669  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6311  df-om 6468  df-1st 6568  df-2nd 6569  df-supp 6682  df-recs 6820  df-rdg 6854  df-1o 6910  df-2o 6911  df-oadd 6914  df-er 7091  df-map 7206  df-pm 7207  df-ixp 7254  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-fsupp 7611  df-fi 7651  df-sup 7681  df-oi 7714  df-card 8099  df-cda 8327  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-div 9984  df-nn 10313  df-2 10370  df-3 10371  df-4 10372  df-5 10373  df-6 10374  df-7 10375  df-8 10376  df-9 10377  df-10 10378  df-n0 10570  df-z 10637  df-dec 10746  df-uz 10852  df-q 10944  df-rp 10982  df-xneg 11079  df-xadd 11080  df-xmul 11081  df-ioo 11294  df-ioc 11295  df-ico 11296  df-icc 11297  df-fz 11427  df-fzo 11535  df-fl 11628  df-mod 11695  df-seq 11793  df-exp 11852  df-fac 12038  df-bc 12065  df-hash 12090  df-shft 12542  df-cj 12574  df-re 12575  df-im 12576  df-sqr 12710  df-abs 12711  df-limsup 12935  df-clim 12952  df-rlim 12953  df-sum 13150  df-ef 13338  df-sin 13340  df-cos 13341  df-pi 13343  df-dvds 13521  df-gcd 13676  df-prm 13749  df-pc 13889  df-struct 14161  df-ndx 14162  df-slot 14163  df-base 14164  df-sets 14165  df-ress 14166  df-plusg 14236  df-mulr 14237  df-starv 14238  df-sca 14239  df-vsca 14240  df-ip 14241  df-tset 14242  df-ple 14243  df-ds 14245  df-unif 14246  df-hom 14247  df-cco 14248  df-rest 14346  df-topn 14347  df-0g 14365  df-gsum 14366  df-topgen 14367  df-pt 14368  df-prds 14371  df-xrs 14425  df-qtop 14430  df-imas 14431  df-xps 14433  df-mre 14509  df-mrc 14510  df-acs 14512  df-mnd 15400  df-submnd 15450  df-mulg 15530  df-cntz 15817  df-cmn 16261  df-psmet 17655  df-xmet 17656  df-met 17657  df-bl 17658  df-mopn 17659  df-fbas 17660  df-fg 17661  df-cnfld 17665  df-top 18347  df-bases 18349  df-topon 18350  df-topsp 18351  df-cld 18467  df-ntr 18468  df-cls 18469  df-nei 18546  df-lp 18584  df-perf 18585  df-cn 18675  df-cnp 18676  df-haus 18763  df-tx 18979  df-hmeo 19172  df-fil 19263  df-fm 19355  df-flim 19356  df-flf 19357  df-xms 19739  df-ms 19740  df-tms 19741  df-cncf 20298  df-limc 21185  df-dv 21186  df-log 21895  df-vma 22322  df-chp 22323
This theorem is referenced by:  pntrlog2bndlem1  22713  pntrlog2bndlem4  22716
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