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Theorem logsqvma2 21225
Description: The Möbius inverse of logsqvma 21224. Equation 10.4.8 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Assertion
Ref Expression
logsqvma2  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Distinct variable group:    x, d, N

Proof of Theorem logsqvma2
Dummy variables  i 
j  k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 11300 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1 ... k )  e. 
Fin )
2 sgmss 20877 . . . . . . . . . 10  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k ) )
3 ssfi 7320 . . . . . . . . . 10  |-  ( ( ( 1 ... k
)  e.  Fin  /\  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k
) )  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
41, 2, 3syl2anc 643 . . . . . . . . 9  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
5 ssrab2 3420 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  k }  C_  NN
6 simpr 448 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  { x  e.  NN  |  x  ||  k } )
75, 6sseldi 3338 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  NN )
8 vmacl 20889 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
97, 8syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  d )  e.  RR )
10 dvdsdivcl 20954 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  { x  e.  NN  |  x  ||  k } )
115, 10sseldi 3338 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  NN )
12 vmacl 20889 . . . . . . . . . . 11  |-  ( ( k  /  d )  e.  NN  ->  (Λ `  ( k  /  d
) )  e.  RR )
1311, 12syl 16 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  ( k  /  d
) )  e.  RR )
149, 13remulcld 9105 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  e.  RR )
154, 14fsumrecl 12516 . . . . . . . 8  |-  ( k  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  e.  RR )
16 vmacl 20889 . . . . . . . . 9  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
17 nnrp 10610 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817relogcld 20506 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( log `  k )  e.  RR )
1916, 18remulcld 9105 . . . . . . . 8  |-  ( k  e.  NN  ->  (
(Λ `  k )  x.  ( log `  k
) )  e.  RR )
2015, 19readdcld 9104 . . . . . . 7  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  RR )
2120recnd 9103 . . . . . 6  |-  ( k  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  CC )
2221adantl 453 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  NN )  ->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) )  e.  CC )
23 eqid 2435 . . . . 5  |-  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )  =  ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) )
2422, 23fmptd 5884 . . . 4  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) : NN --> CC )
25 ssrab2 3420 . . . . . . . . 9  |-  { x  e.  NN  |  x  ||  n }  C_  NN
26 simpr 448 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  {
x  e.  NN  |  x  ||  n } )
2725, 26sseldi 3338 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  NN )
28 breq2 4208 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
x  ||  k  <->  x  ||  m
) )
2928rabbidv 2940 . . . . . . . . . . 11  |-  ( k  =  m  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  m } )
30 oveq1 6079 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  (
k  /  d )  =  ( m  / 
d ) )
3130fveq2d 5723 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( m  /  d
) ) )
3231oveq2d 6088 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3332adantr 452 . . . . . . . . . . 11  |-  ( ( k  =  m  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3429, 33sumeq12dv 12488 . . . . . . . . . 10  |-  ( k  =  m  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
35 fveq2 5719 . . . . . . . . . . 11  |-  ( k  =  m  ->  (Λ `  k )  =  (Λ `  m ) )
36 fveq2 5719 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( log `  k )  =  ( log `  m
) )
3735, 36oveq12d 6090 . . . . . . . . . 10  |-  ( k  =  m  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  m )  x.  ( log `  m
) ) )
3834, 37oveq12d 6090 . . . . . . . . 9  |-  ( k  =  m  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
39 ovex 6097 . . . . . . . . 9  |-  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  e.  _V
4038, 23, 39fvmpt3i 5800 . . . . . . . 8  |-  ( m  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) ) )
4127, 40syl 16 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  ( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) ) ) `  m )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  m } 
( (Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
4241sumeq2dv 12485 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
)  =  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) ) )
43 logsqvma 21224 . . . . . . 7  |-  ( n  e.  NN  ->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( sum_ d  e.  {
x  e.  NN  |  x  ||  m }  (
(Λ `  d )  x.  (Λ `  ( m  /  d ) ) )  +  ( (Λ `  m )  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4443adantl 453 . . . . . 6  |-  ( ( N  e.  NN  /\  n  e.  NN )  -> 
sum_ m  e.  { x  e.  NN  |  x  ||  n }  ( sum_ d  e.  { x  e.  NN  |  x  ||  m }  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) )  +  ( (Λ `  m
)  x.  ( log `  m ) ) )  =  ( ( log `  n ) ^ 2 ) )
4542, 44eqtr2d 2468 . . . . 5  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( log `  n
) ^ 2 )  =  sum_ m  e.  {
x  e.  NN  |  x  ||  n }  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  m
) )
4645mpteq2dva 4287 . . . 4  |-  ( N  e.  NN  ->  (
n  e.  NN  |->  ( ( log `  n
) ^ 2 ) )  =  ( n  e.  NN  |->  sum_ m  e.  { x  e.  NN  |  x  ||  n } 
( ( k  e.  NN  |->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k
) ) ) ) `
 m ) ) )
4724, 46muinv 20966 . . 3  |-  ( N  e.  NN  ->  (
k  e.  NN  |->  (
sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) )
4847fveq1d 5721 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N ) )
49 breq2 4208 . . . . . 6  |-  ( k  =  N  ->  (
x  ||  k  <->  x  ||  N
) )
5049rabbidv 2940 . . . . 5  |-  ( k  =  N  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  N } )
51 oveq1 6079 . . . . . . . 8  |-  ( k  =  N  ->  (
k  /  d )  =  ( N  / 
d ) )
5251fveq2d 5723 . . . . . . 7  |-  ( k  =  N  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( N  /  d
) ) )
5352oveq2d 6088 . . . . . 6  |-  ( k  =  N  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5453adantr 452 . . . . 5  |-  ( ( k  =  N  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5550, 54sumeq12dv 12488 . . . 4  |-  ( k  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
56 fveq2 5719 . . . . 5  |-  ( k  =  N  ->  (Λ `  k )  =  (Λ `  N ) )
57 fveq2 5719 . . . . 5  |-  ( k  =  N  ->  ( log `  k )  =  ( log `  N
) )
5856, 57oveq12d 6090 . . . 4  |-  ( k  =  N  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  N )  x.  ( log `  N
) ) )
5955, 58oveq12d 6090 . . 3  |-  ( k  =  N  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d )  x.  (Λ `  ( k  /  d
) ) )  +  ( (Λ `  k
)  x.  ( log `  k ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
6059, 23, 39fvmpt3i 5800 . 2  |-  ( N  e.  NN  ->  (
( k  e.  NN  |->  ( sum_ d  e.  {
x  e.  NN  |  x  ||  k }  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  +  ( (Λ `  k )  x.  ( log `  k ) ) ) ) `  N
)  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) )  +  ( (Λ `  N
)  x.  ( log `  N ) ) ) )
61 fveq2 5719 . . . . . 6  |-  ( j  =  d  ->  (
mmu `  j )  =  ( mmu `  d ) )
62 oveq2 6080 . . . . . . . 8  |-  ( j  =  d  ->  (
i  /  j )  =  ( i  / 
d ) )
6362fveq2d 5723 . . . . . . 7  |-  ( j  =  d  ->  ( log `  ( i  / 
j ) )  =  ( log `  (
i  /  d ) ) )
6463oveq1d 6087 . . . . . 6  |-  ( j  =  d  ->  (
( log `  (
i  /  j ) ) ^ 2 )  =  ( ( log `  ( i  /  d
) ) ^ 2 ) )
6561, 64oveq12d 6090 . . . . 5  |-  ( j  =  d  ->  (
( mmu `  j
)  x.  ( ( log `  ( i  /  j ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) ) )
6665cbvsumv 12478 . . . 4  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )
67 breq2 4208 . . . . . 6  |-  ( i  =  N  ->  (
x  ||  i  <->  x  ||  N
) )
6867rabbidv 2940 . . . . 5  |-  ( i  =  N  ->  { x  e.  NN  |  x  ||  i }  =  {
x  e.  NN  |  x  ||  N } )
69 oveq1 6079 . . . . . . . . 9  |-  ( i  =  N  ->  (
i  /  d )  =  ( N  / 
d ) )
7069fveq2d 5723 . . . . . . . 8  |-  ( i  =  N  ->  ( log `  ( i  / 
d ) )  =  ( log `  ( N  /  d ) ) )
7170oveq1d 6087 . . . . . . 7  |-  ( i  =  N  ->  (
( log `  (
i  /  d ) ) ^ 2 )  =  ( ( log `  ( N  /  d
) ) ^ 2 ) )
7271oveq2d 6088 . . . . . 6  |-  ( i  =  N  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7372adantr 452 . . . . 5  |-  ( ( i  =  N  /\  d  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7468, 73sumeq12dv 12488 . . . 4  |-  ( i  =  N  ->  sum_ d  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7566, 74syl5eq 2479 . . 3  |-  ( i  =  N  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
76 ssrab2 3420 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  i }  C_  NN
77 dvdsdivcl 20954 . . . . . . . 8  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  { x  e.  NN  |  x  ||  i } )
7876, 77sseldi 3338 . . . . . . 7  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  NN )
79 fveq2 5719 . . . . . . . . 9  |-  ( n  =  ( i  / 
j )  ->  ( log `  n )  =  ( log `  (
i  /  j ) ) )
8079oveq1d 6087 . . . . . . . 8  |-  ( n  =  ( i  / 
j )  ->  (
( log `  n
) ^ 2 )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
81 eqid 2435 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )  =  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )
82 ovex 6097 . . . . . . . 8  |-  ( ( log `  n ) ^ 2 )  e. 
_V
8380, 81, 82fvmpt3i 5800 . . . . . . 7  |-  ( ( i  /  j )  e.  NN  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8478, 83syl 16 . . . . . 6  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8584oveq2d 6088 . . . . 5  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( mmu `  j
)  x.  ( ( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  ( ( mmu `  j )  x.  ( ( log `  ( i  /  j
) ) ^ 2 ) ) )
8685sumeq2dv 12485 . . . 4  |-  ( i  e.  NN  ->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) )  =  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) ) )
8786mpteq2ia 4283 . . 3  |-  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) ) ) )  =  ( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( log `  (
i  /  j ) ) ^ 2 ) ) )
88 sumex 12469 . . 3  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  e.  _V
8975, 87, 88fvmpt3i 5800 . 2  |-  ( N  e.  NN  ->  (
( i  e.  NN  |->  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( (
mmu `  j )  x.  ( ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) ) `  (
i  /  j ) ) ) ) `  N )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (
mmu `  d )  x.  ( ( log `  ( N  /  d ) ) ^ 2 ) ) )
9048, 60, 893eqtr3rd 2476 1  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  (Λ `  ( N  /  d ) ) )  +  ( (Λ `  N )  x.  ( log `  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701    C_ wss 3312   class class class wbr 4204    e. cmpt 4258   ` cfv 5445  (class class class)co 6072   Fincfn 7100   CCcc 8977   RRcr 8978   1c1 8980    + caddc 8982    x. cmul 8984    / cdiv 9666   NNcn 9989   2c2 10038   ...cfz 11032   ^cexp 11370   sum_csu 12467    || cdivides 12840   logclog 20440  Λcvma 20862   mmucmu 20865
This theorem is referenced by:  selberg  21230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ioc 10910  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-fl 11190  df-mod 11239  df-seq 11312  df-exp 11371  df-fac 11555  df-bc 11582  df-hash 11607  df-shft 11870  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-limsup 12253  df-clim 12270  df-rlim 12271  df-sum 12468  df-ef 12658  df-sin 12660  df-cos 12661  df-pi 12663  df-dvds 12841  df-gcd 12995  df-prm 13068  df-pc 13199  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742  df-log 20442  df-vma 20868  df-mu 20871
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