MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pntsval2 Unicode version

Theorem pntsval2 20837
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 20833 . 2  |-  ( A  e.  RR  ->  ( S `  A )  =  sum_ n  e.  ( 1 ... ( |_
`  A ) ) ( (Λ `  n
)  x.  ( ( log `  n )  +  (ψ `  ( A  /  n ) ) ) ) )
3 elfznn 10911 . . . . . . 7  |-  ( n  e.  ( 1 ... ( |_ `  A
) )  ->  n  e.  NN )
43adantl 452 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  NN )
5 vmacl 20468 . . . . . 6  |-  ( n  e.  NN  ->  (Λ `  n )  e.  RR )
64, 5syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  RR )
76recnd 8951 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  n )  e.  CC )
84nnrpd 10481 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  n  e.  RR+ )
98relogcld 20085 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  RR )
109recnd 8951 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( log `  n
)  e.  CC )
11 simpl 443 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
1211, 4nndivred 9884 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  n )  e.  RR )
13 chpcl 20474 . . . . . 6  |-  ( ( A  /  n )  e.  RR  ->  (ψ `  ( A  /  n
) )  e.  RR )
1412, 13syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  RR )
1514recnd 8951 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  n ) )  e.  CC )
167, 10, 15adddid 8949 . . 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 12273 . 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 5608 . . . . . . 7  |-  ( n  =  m  ->  (Λ `  n )  =  (Λ `  m ) )
19 oveq2 5953 . . . . . . . 8  |-  ( n  =  m  ->  ( A  /  n )  =  ( A  /  m
) )
2019fveq2d 5612 . . . . . . 7  |-  ( n  =  m  ->  (ψ `  ( A  /  n
) )  =  (ψ `  ( A  /  m
) ) )
2118, 20oveq12d 5963 . . . . . 6  |-  ( n  =  m  ->  (
(Λ `  n )  x.  (ψ `  ( A  /  n ) ) )  =  ( (Λ `  m
)  x.  (ψ `  ( A  /  m
) ) ) )
2221cbvsumv 12266 . . . . 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 11127 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... ( |_ `  ( A  /  m ) ) )  e.  Fin )
24 elfznn 10911 . . . . . . . . . . . 12  |-  ( m  e.  ( 1 ... ( |_ `  A
) )  ->  m  e.  NN )
2524adantl 452 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  m  e.  NN )
26 vmacl 20468 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  (Λ `  m )  e.  RR )
2725, 26syl 15 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  RR )
2827recnd 8951 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (Λ `  m )  e.  CC )
29 elfznn 10911 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( |_ `  ( A  /  m ) ) )  ->  k  e.  NN )
3029adantl 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  NN )
31 vmacl 20468 . . . . . . . . . . 11  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
3230, 31syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  RR )
3332recnd 8951 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  k
)  e.  CC )
3423, 28, 33fsummulc2 12343 . . . . . . . 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 443 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  A  e.  RR )
3635, 25nndivred 9884 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( A  /  m )  e.  RR )
37 chpval 20472 . . . . . . . . . 10  |-  ( ( A  /  m )  e.  RR  ->  (ψ `  ( A  /  m
) )  =  sum_ k  e.  ( 1 ... ( |_ `  ( A  /  m
) ) ) (Λ `  k ) )
3836, 37syl 15 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  ->  (ψ `  ( A  /  m ) )  =  sum_ k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) (Λ `  k )
)
3938oveq2d 5961 . . . . . . . 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 9852 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  k  e.  CC )
4124ad2antlr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  NN )
4241nncnd 9852 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  e.  CC )
4341nnne0d 9880 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  m  =/=  0 )
4440, 42, 43divcan3d 9631 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  ( (
m  x.  k )  /  m )  =  k )
4544fveq2d 5612 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  m  e.  ( 1 ... ( |_ `  A ) ) )  /\  k  e.  ( 1 ... ( |_
`  ( A  /  m ) ) ) )  ->  (Λ `  (
( m  x.  k
)  /  m ) )  =  (Λ `  k
) )
4645oveq2d 5961 . . . . . . . . 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 12273 . . . . . . . 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 2400 . . . . . . 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 12273 . . . . . 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 5952 . . . . . . . . 9  |-  ( n  =  ( m  x.  k )  ->  (
n  /  m )  =  ( ( m  x.  k )  /  m ) )
5150fveq2d 5612 . . . . . . . 8  |-  ( n  =  ( m  x.  k )  ->  (Λ `  ( n  /  m
) )  =  (Λ `  ( ( m  x.  k )  /  m
) ) )
5251oveq2d 5961 . . . . . . 7  |-  ( n  =  ( m  x.  k )  ->  (
(Λ `  m )  x.  (Λ `  ( n  /  m ) ) )  =  ( (Λ `  m
)  x.  (Λ `  (
( m  x.  k
)  /  m ) ) ) )
53 id 19 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  RR )
54 ssrab2 3334 . . . . . . . . . . . 12  |-  { y  e.  NN  |  y 
||  n }  C_  NN
55 simpr 447 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  { y  e.  NN  | 
y  ||  n }
)
5654, 55sseldi 3254 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  m  e.  NN )
5756, 26syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  m
)  e.  RR )
58 dvdsdivcl 20533 . . . . . . . . . . . . 13  |-  ( ( n  e.  NN  /\  m  e.  { y  e.  NN  |  y  ||  n } )  ->  (
n  /  m )  e.  { y  e.  NN  |  y  ||  n } )
594, 58sylan 457 . . . . . . . . . . . 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 3254 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( n  /  m )  e.  NN )
61 vmacl 20468 . . . . . . . . . . 11  |-  ( ( n  /  m )  e.  NN  ->  (Λ `  ( n  /  m
) )  e.  RR )
6260, 61syl 15 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  (Λ `  (
n  /  m ) )  e.  RR )
6357, 62remulcld 8953 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  RR )
6463recnd 8951 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  /\  m  e.  {
y  e.  NN  | 
y  ||  n }
)  ->  ( (Λ `  m )  x.  (Λ `  ( n  /  m
) ) )  e.  CC )
6564anasss 628 . . . . . . 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 20540 . . . . . 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 2393 . . . . 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 2402 . . . 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 5961 . . 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 11127 . . . 4  |-  ( A  e.  RR  ->  (
1 ... ( |_ `  A ) )  e. 
Fin )
717, 10mulcld 8945 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  ( log `  n ) )  e.  CC )
727, 15mulcld 8945 . . . 4  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( (Λ `  n
)  x.  (ψ `  ( A  /  n
) ) )  e.  CC )
7370, 71, 72fsumadd 12308 . . 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 11127 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  ( 1 ... n )  e.  Fin )
75 sgmss 20456 . . . . . . . 8  |-  ( n  e.  NN  ->  { y  e.  NN  |  y 
||  n }  C_  ( 1 ... n
) )
764, 75syl 15 . . . . . . 7  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  C_  ( 1 ... n ) )
77 ssfi 7171 . . . . . . 7  |-  ( ( ( 1 ... n
)  e.  Fin  /\  { y  e.  NN  | 
y  ||  n }  C_  ( 1 ... n
) )  ->  { y  e.  NN  |  y 
||  n }  e.  Fin )
7874, 76, 77syl2anc 642 . . . . . 6  |-  ( ( A  e.  RR  /\  n  e.  ( 1 ... ( |_ `  A ) ) )  ->  { y  e.  NN  |  y  ||  n }  e.  Fin )
7978, 63fsumrecl 12304 . . . . 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 8951 . . . 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 12308 . . 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 2400 . 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 2394 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 358    = wceq 1642    e. wcel 1710   {crab 2623    C_ wss 3228   class class class wbr 4104    e. cmpt 4158   ` cfv 5337  (class class class)co 5945   Fincfn 6951   CCcc 8825   RRcr 8826   1c1 8828    + caddc 8830    x. cmul 8832    / cdiv 9513   NNcn 9836   ...cfz 10874   |_cfl 11016   sum_csu 12255    || cdivides 12628   logclog 20019  Λcvma 20441  ψcchp 20442
This theorem is referenced by:  pntrlog2bndlem1  20838  pntrlog2bndlem4  20841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-2o 6567  df-oadd 6570  df-er 6747  df-map 6862  df-pm 6863  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-fi 7255  df-sup 7284  df-oi 7315  df-card 7662  df-cda 7884  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-uz 10323  df-q 10409  df-rp 10447  df-xneg 10544  df-xadd 10545  df-xmul 10546  df-ioo 10752  df-ioc 10753  df-ico 10754  df-icc 10755  df-fz 10875  df-fzo 10963  df-fl 11017  df-mod 11066  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446  df-sin 12448  df-cos 12449  df-pi 12451  df-dvds 12629  df-gcd 12783  df-prm 12856  df-pc 12987  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-starv 13320  df-sca 13321  df-vsca 13322  df-tset 13324  df-ple 13325  df-ds 13327  df-unif 13328  df-hom 13329  df-cco 13330  df-rest 13426  df-topn 13427  df-topgen 13443  df-pt 13444  df-prds 13447  df-xrs 13502  df-0g 13503  df-gsum 13504  df-qtop 13509  df-imas 13510  df-xps 13512  df-mre 13587  df-mrc 13588  df-acs 13590  df-mnd 14466  df-submnd 14515  df-mulg 14591  df-cntz 14892  df-cmn 15190  df-xmet 16475  df-met 16476  df-bl 16477  df-mopn 16478  df-fbas 16479  df-fg 16480  df-cnfld 16483  df-top 16742  df-bases 16744  df-topon 16745  df-topsp 16746  df-cld 16862  df-ntr 16863  df-cls 16864  df-nei 16941  df-lp 16974  df-perf 16975  df-cn 17063  df-cnp 17064  df-haus 17149  df-tx 17363  df-hmeo 17552  df-fil 17643  df-fm 17735  df-flim 17736  df-flf 17737  df-xms 17987  df-ms 17988  df-tms 17989  df-cncf 18485  df-limc 19320  df-dv 19321  df-log 20021  df-vma 20447  df-chp 20448
  Copyright terms: Public domain W3C validator