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Theorem pntsval2 21301
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 21297 . 2  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
3 elfznn 11111 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 454 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 vmacl 20932 . . . . . 6  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
64, 5syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
76recnd 9145 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
84nnrpd 10678 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  RR+ )
98relogcld 20549 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  RR )
109recnd 9145 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  CC )
11 simpl 445 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
1211, 4nndivred 10079 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  n )  e.  RR )
13 chpcl 20938 . . . . . 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 9145 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  CC )
167, 10, 15adddid 9143 . . 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 12528 . 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 5757 . . . . . . 7  |-  ( n  =  m  ->  (Λ `  n )  =  (Λ `  m ) )
19 oveq2 6118 . . . . . . . 8  |-  ( n  =  m  ->  ( A  /  n )  =  ( A  /  m
) )
2019fveq2d 5761 . . . . . . 7  |-  ( n  =  m  ->  (ψ `  ( A  /  n
) )  =  (ψ `  ( A  /  m
) ) )
2118, 20oveq12d 6128 . . . . . 6  |-  ( n  =  m  ->  (
(Λ `  n )  x.  (ψ `  ( A  /  n ) ) )  =  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) ) )
2221cbvsumv 12521 . . . . 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 11343 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  m ) ) )  e.  Fin )
24 elfznn 11111 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
2524adantl 454 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
26 vmacl 20932 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
2725, 26syl 16 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  RR )
2827recnd 9145 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  CC )
29 elfznn 11111 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) )  ->  k  e.  NN )
3029adantl 454 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  NN )
31 vmacl 20932 . . . . . . . . . . 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 9145 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  CC )
3423, 28, 33fsummulc2 12598 . . . . . . . 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 445 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
3635, 25nndivred 10079 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR )
37 chpval 20936 . . . . . . . . . 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 6126 . . . . . . . 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 10047 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  CC )
4124ad2antlr 709 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  NN )
4241nncnd 10047 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  CC )
4341nnne0d 10075 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  =/=  0 )
4440, 42, 43divcan3d 9826 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  ( (
m  x.  k )  /  m )  =  k )
4544fveq2d 5761 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  (
( m  x.  k
)  /  m ) )  =  (Λ `  k
) )
4645oveq2d 6126 . . . . . . . . 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 12528 . . . . . . . 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 2484 . . . . . . 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 12528 . . . . . 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 6117 . . . . . . . . 9  |-  ( n  =  ( m  x.  k )  ->  (
n  /  m )  =  ( ( m  x.  k )  /  m ) )
5150fveq2d 5761 . . . . . . . 8  |-  ( n  =  ( m  x.  k )  ->  (Λ `  ( n  /  m
) )  =  (Λ `  ( ( m  x.  k )  /  m
) ) )
5251oveq2d 6126 . . . . . . 7  |-  ( n  =  ( m  x.  k )  ->  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  =  ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
53 id 21 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR )
54 ssrab2 3414 . . . . . . . . . . . 12  |-  { y  e.  NN  |  y 
||  n }  C_  NN
55 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  { y  e.  NN  | 
y  ||  n }
)
5654, 55sseldi 3332 . . . . . . . . . . 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 20997 . . . . . . . . . . . . 13  |-  ( ( n  e.  NN  /\  m  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  m )  e.  { y  e.  NN  |  y  ||  n } )
594, 58sylan 459 . . . . . . . . . . . 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 3332 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  m )  e.  NN )
61 vmacl 20932 . . . . . . . . . . 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 9147 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  RR )
6463recnd 9145 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  CC )
6564anasss 630 . . . . . . 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 21004 . . . . . 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 2477 . . . . 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 2486 . . . 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 6126 . . 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 11343 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
717, 10mulcld 9139 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  ( log `  n ) )  e.  CC )
727, 15mulcld 9139 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  e.  CC )
7370, 71, 72fsumadd 12563 . . 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 11343 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e.  Fin )
75 sgmss 20920 . . . . . . . 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 7358 . . . . . . 7  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
7874, 76, 77syl2anc 644 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
7978, 63fsumrecl 12559 . . . . 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 9145 . . . 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 12563 . . 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 2484 . 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 2478 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 360    = wceq 1653    e. wcel 1727   {crab 2715    C_ wss 3306   class class class wbr 4237    e. cmpt 4291   ` cfv 5483  (class class class)co 6110   Fincfn 7138   CCcc 9019   RRcr 9020   1c1 9022    + caddc 9024    x. cmul 9026    / cdiv 9708   NNcn 10031   ...cfz 11074   |_cfl 11232   sum_csu 12510    || cdivides 12883   logclog 20483  Λcvma 20905  ψcchp 20906
This theorem is referenced by:  pntrlog2bndlem1  21302  pntrlog2bndlem4  21305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098  ax-pre-sup 9099  ax-addf 9100  ax-mulf 9101
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-iin 4120  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-pm 7050  df-ixp 7093  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-fi 7445  df-sup 7475  df-oi 7508  df-card 7857  df-cda 8079  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-10 10097  df-n0 10253  df-z 10314  df-dec 10414  df-uz 10520  df-q 10606  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ioo 10951  df-ioc 10952  df-ico 10953  df-icc 10954  df-fz 11075  df-fzo 11167  df-fl 11233  df-mod 11282  df-seq 11355  df-exp 11414  df-fac 11598  df-bc 11625  df-hash 11650  df-shft 11913  df-cj 11935  df-re 11936  df-im 11937  df-sqr 12071  df-abs 12072  df-limsup 12296  df-clim 12313  df-rlim 12314  df-sum 12511  df-ef 12701  df-sin 12703  df-cos 12704  df-pi 12706  df-dvds 12884  df-gcd 13038  df-prm 13111  df-pc 13242  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-starv 13575  df-sca 13576  df-vsca 13577  df-tset 13579  df-ple 13580  df-ds 13582  df-unif 13583  df-hom 13584  df-cco 13585  df-rest 13681  df-topn 13682  df-topgen 13698  df-pt 13699  df-prds 13702  df-xrs 13757  df-0g 13758  df-gsum 13759  df-qtop 13764  df-imas 13765  df-xps 13767  df-mre 13842  df-mrc 13843  df-acs 13845  df-mnd 14721  df-submnd 14770  df-mulg 14846  df-cntz 15147  df-cmn 15445  df-psmet 16725  df-xmet 16726  df-met 16727  df-bl 16728  df-mopn 16729  df-fbas 16730  df-fg 16731  df-cnfld 16735  df-top 16994  df-bases 16996  df-topon 16997  df-topsp 16998  df-cld 17114  df-ntr 17115  df-cls 17116  df-nei 17193  df-lp 17231  df-perf 17232  df-cn 17322  df-cnp 17323  df-haus 17410  df-tx 17625  df-hmeo 17818  df-fil 17909  df-fm 18001  df-flim 18002  df-flf 18003  df-xms 18381  df-ms 18382  df-tms 18383  df-cncf 18939  df-limc 19784  df-dv 19785  df-log 20485  df-vma 20911  df-chp 20912
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