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Theorem pntsval2 21227
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 21223 . 2  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
3 elfznn 11040 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 vmacl 20858 . . . . . 6  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
64, 5syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
76recnd 9074 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
84nnrpd 10607 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  RR+ )
98relogcld 20475 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  RR )
109recnd 9074 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  CC )
11 simpl 444 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
1211, 4nndivred 10008 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  n )  e.  RR )
13 chpcl 20864 . . . . . 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 9074 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  CC )
167, 10, 15adddid 9072 . . 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 12456 . 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 5691 . . . . . . 7  |-  ( n  =  m  ->  (Λ `  n )  =  (Λ `  m ) )
19 oveq2 6052 . . . . . . . 8  |-  ( n  =  m  ->  ( A  /  n )  =  ( A  /  m
) )
2019fveq2d 5695 . . . . . . 7  |-  ( n  =  m  ->  (ψ `  ( A  /  n
) )  =  (ψ `  ( A  /  m
) ) )
2118, 20oveq12d 6062 . . . . . 6  |-  ( n  =  m  ->  (
(Λ `  n )  x.  (ψ `  ( A  /  n ) ) )  =  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) ) )
2221cbvsumv 12449 . . . . 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 11271 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  m ) ) )  e.  Fin )
24 elfznn 11040 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
2524adantl 453 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
26 vmacl 20858 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
2725, 26syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  RR )
2827recnd 9074 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  CC )
29 elfznn 11040 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) )  ->  k  e.  NN )
3029adantl 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  NN )
31 vmacl 20858 . . . . . . . . . . 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 9074 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  CC )
3423, 28, 33fsummulc2 12526 . . . . . . . 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 444 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
3635, 25nndivred 10008 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR )
37 chpval 20862 . . . . . . . . . 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 6060 . . . . . . . 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 9976 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  CC )
4124ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  NN )
4241nncnd 9976 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  CC )
4341nnne0d 10004 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  =/=  0 )
4440, 42, 43divcan3d 9755 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  ( (
m  x.  k )  /  m )  =  k )
4544fveq2d 5695 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  (
( m  x.  k
)  /  m ) )  =  (Λ `  k
) )
4645oveq2d 6060 . . . . . . . . 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 12456 . . . . . . . 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 2450 . . . . . . 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 12456 . . . . . 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 6051 . . . . . . . . 9  |-  ( n  =  ( m  x.  k )  ->  (
n  /  m )  =  ( ( m  x.  k )  /  m ) )
5150fveq2d 5695 . . . . . . . 8  |-  ( n  =  ( m  x.  k )  ->  (Λ `  ( n  /  m
) )  =  (Λ `  ( ( m  x.  k )  /  m
) ) )
5251oveq2d 6060 . . . . . . 7  |-  ( n  =  ( m  x.  k )  ->  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  =  ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
53 id 20 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR )
54 ssrab2 3392 . . . . . . . . . . . 12  |-  { y  e.  NN  |  y 
||  n }  C_  NN
55 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  { y  e.  NN  | 
y  ||  n }
)
5654, 55sseldi 3310 . . . . . . . . . . 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 20923 . . . . . . . . . . . . 13  |-  ( ( n  e.  NN  /\  m  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  m )  e.  { y  e.  NN  |  y  ||  n } )
594, 58sylan 458 . . . . . . . . . . . 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 3310 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  m )  e.  NN )
61 vmacl 20858 . . . . . . . . . . 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 9076 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  RR )
6463recnd 9074 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  CC )
6564anasss 629 . . . . . . 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 20930 . . . . . 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 2443 . . . . 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 2452 . . . 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 6060 . . 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 11271 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
717, 10mulcld 9068 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  ( log `  n ) )  e.  CC )
727, 15mulcld 9068 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  e.  CC )
7370, 71, 72fsumadd 12491 . . 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 11271 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e.  Fin )
75 sgmss 20846 . . . . . . . 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 7292 . . . . . . 7  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
7874, 76, 77syl2anc 643 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
7978, 63fsumrecl 12487 . . . . 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 9074 . . . 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 12491 . . 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 2450 . 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 2444 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 set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674    C_ wss 3284   class class class wbr 4176    e. cmpt 4230   ` cfv 5417  (class class class)co 6044   Fincfn 7072   CCcc 8948   RRcr 8949   1c1 8951    + caddc 8953    x. cmul 8955    / cdiv 9637   NNcn 9960   ...cfz 11003   |_cfl 11160   sum_csu 12438    || cdivides 12811   logclog 20409  Λcvma 20831  ψcchp 20832
This theorem is referenced by:  pntrlog2bndlem1  21228  pntrlog2bndlem4  21231
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029  ax-mulf 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-ixp 7027  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-oi 7439  df-card 7786  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ioo 10880  df-ioc 10881  df-ico 10882  df-icc 10883  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-fac 11526  df-bc 11553  df-hash 11578  df-shft 11841  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-limsup 12224  df-clim 12241  df-rlim 12242  df-sum 12439  df-ef 12629  df-sin 12631  df-cos 12632  df-pi 12634  df-dvds 12812  df-gcd 12966  df-prm 13039  df-pc 13170  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-starv 13503  df-sca 13504  df-vsca 13505  df-tset 13507  df-ple 13508  df-ds 13510  df-unif 13511  df-hom 13512  df-cco 13513  df-rest 13609  df-topn 13610  df-topgen 13626  df-pt 13627  df-prds 13630  df-xrs 13685  df-0g 13686  df-gsum 13687  df-qtop 13692  df-imas 13693  df-xps 13695  df-mre 13770  df-mrc 13771  df-acs 13773  df-mnd 14649  df-submnd 14698  df-mulg 14774  df-cntz 15075  df-cmn 15373  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-fbas 16658  df-fg 16659  df-cnfld 16663  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-lp 17159  df-perf 17160  df-cn 17249  df-cnp 17250  df-haus 17337  df-tx 17551  df-hmeo 17744  df-fil 17835  df-fm 17927  df-flim 17928  df-flf 17929  df-xms 18307  df-ms 18308  df-tms 18309  df-cncf 18865  df-limc 19710  df-dv 19711  df-log 20411  df-vma 20837  df-chp 20838
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