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Theorem logsqvma2 20654
Description: The Möbius inverse of logsqvma 20653. 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
StepHypRef Expression
1 fzfid 11001 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1 ... k )  e. 
Fin )
2 sgmss 20306 . . . . . . . . . 10  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k ) )
3 ssfi 7051 . . . . . . . . . 10  |-  ( ( ( 1 ... k
)  e.  Fin  /\  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k
) )  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
41, 2, 3syl2anc 645 . . . . . . . . 9  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
5 ssrab2 3233 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  k }  C_  NN
6 simpr 449 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  { x  e.  NN  |  x  ||  k } )
75, 6sseldi 3153 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  NN )
8 vmacl 20318 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
97, 8syl 17 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  d )  e.  RR )
10 dvdsdivcl 20383 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  { x  e.  NN  |  x  ||  k } )
115, 10sseldi 3153 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  NN )
12 vmacl 20318 . . . . . . . . . . 11  |-  ( ( k  /  d )  e.  NN  ->  (Λ `  ( k  /  d
) )  e.  RR )
1311, 12syl 17 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  ( k  /  d
) )  e.  RR )
149, 13remulcld 8831 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  e.  RR )
154, 14fsumrecl 12172 . . . . . . . 8  |-  ( k  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  e.  RR )
16 vmacl 20318 . . . . . . . . 9  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
17 nnrp 10330 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817relogcld 19936 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( log `  k )  e.  RR )
1916, 18remulcld 8831 . . . . . . . 8  |-  ( k  e.  NN  ->  (
(Λ `  k )  x.  ( log `  k
) )  e.  RR )
2015, 19readdcld 8830 . . . . . . 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 8829 . . . . . 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 454 . . . . 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 2258 . . . . 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 5618 . . . 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 3233 . . . . . . . . 9  |-  { x  e.  NN  |  x  ||  n }  C_  NN
26 simpr 449 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  {
x  e.  NN  |  x  ||  n } )
2725, 26sseldi 3153 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  NN )
28 breq2 4001 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
x  ||  k  <->  x  ||  m
) )
2928rabbidv 2755 . . . . . . . . . . 11  |-  ( k  =  m  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  m } )
30 oveq1 5799 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  (
k  /  d )  =  ( m  / 
d ) )
3130fveq2d 5462 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( m  /  d
) ) )
3231oveq2d 5808 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3332adantr 453 . . . . . . . . . . 11  |-  ( ( k  =  m  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3429, 33sumeq12dv 12144 . . . . . . . . . 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 5458 . . . . . . . . . . 11  |-  ( k  =  m  ->  (Λ `  k )  =  (Λ `  m ) )
36 fveq2 5458 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( log `  k )  =  ( log `  m
) )
3735, 36oveq12d 5810 . . . . . . . . . 10  |-  ( k  =  m  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  m )  x.  ( log `  m
) ) )
3834, 37oveq12d 5810 . . . . . . . . 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 5817 . . . . . . . . 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 5539 . . . . . . . 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 17 . . . . . . 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 12141 . . . . . 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 20653 . . . . . . 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 454 . . . . . 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 2291 . . . . 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 4080 . . . 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 20395 . . 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 5460 . 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 4001 . . . . . 6  |-  ( k  =  N  ->  (
x  ||  k  <->  x  ||  N
) )
5049rabbidv 2755 . . . . 5  |-  ( k  =  N  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  N } )
51 oveq1 5799 . . . . . . . 8  |-  ( k  =  N  ->  (
k  /  d )  =  ( N  / 
d ) )
5251fveq2d 5462 . . . . . . 7  |-  ( k  =  N  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( N  /  d
) ) )
5352oveq2d 5808 . . . . . 6  |-  ( k  =  N  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5453adantr 453 . . . . 5  |-  ( ( k  =  N  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5550, 54sumeq12dv 12144 . . . 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 5458 . . . . 5  |-  ( k  =  N  ->  (Λ `  k )  =  (Λ `  N ) )
57 fveq2 5458 . . . . 5  |-  ( k  =  N  ->  ( log `  k )  =  ( log `  N
) )
5856, 57oveq12d 5810 . . . 4  |-  ( k  =  N  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  N )  x.  ( log `  N
) ) )
5955, 58oveq12d 5810 . . 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 5539 . 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 5458 . . . . . 6  |-  ( j  =  d  ->  (
mmu `  j )  =  ( mmu `  d ) )
62 oveq2 5800 . . . . . . . 8  |-  ( j  =  d  ->  (
i  /  j )  =  ( i  / 
d ) )
6362fveq2d 5462 . . . . . . 7  |-  ( j  =  d  ->  ( log `  ( i  / 
j ) )  =  ( log `  (
i  /  d ) ) )
6463oveq1d 5807 . . . . . 6  |-  ( j  =  d  ->  (
( log `  (
i  /  j ) ) ^ 2 )  =  ( ( log `  ( i  /  d
) ) ^ 2 ) )
6561, 64oveq12d 5810 . . . . 5  |-  ( j  =  d  ->  (
( mmu `  j
)  x.  ( ( log `  ( i  /  j ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) ) )
6665cbvsumv 12134 . . . 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 4001 . . . . . 6  |-  ( i  =  N  ->  (
x  ||  i  <->  x  ||  N
) )
6867rabbidv 2755 . . . . 5  |-  ( i  =  N  ->  { x  e.  NN  |  x  ||  i }  =  {
x  e.  NN  |  x  ||  N } )
69 oveq1 5799 . . . . . . . . 9  |-  ( i  =  N  ->  (
i  /  d )  =  ( N  / 
d ) )
7069fveq2d 5462 . . . . . . . 8  |-  ( i  =  N  ->  ( log `  ( i  / 
d ) )  =  ( log `  ( N  /  d ) ) )
7170oveq1d 5807 . . . . . . 7  |-  ( i  =  N  ->  (
( log `  (
i  /  d ) ) ^ 2 )  =  ( ( log `  ( N  /  d
) ) ^ 2 ) )
7271oveq2d 5808 . . . . . 6  |-  ( i  =  N  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7372adantr 453 . . . . 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 12144 . . . 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 2302 . . 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 3233 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  i }  C_  NN
77 dvdsdivcl 20383 . . . . . . . 8  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  { x  e.  NN  |  x  ||  i } )
7876, 77sseldi 3153 . . . . . . 7  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  NN )
79 fveq2 5458 . . . . . . . . 9  |-  ( n  =  ( i  / 
j )  ->  ( log `  n )  =  ( log `  (
i  /  j ) ) )
8079oveq1d 5807 . . . . . . . 8  |-  ( n  =  ( i  / 
j )  ->  (
( log `  n
) ^ 2 )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
81 eqid 2258 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )  =  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )
82 ovex 5817 . . . . . . . 8  |-  ( ( log `  n ) ^ 2 )  e. 
_V
8380, 81, 82fvmpt3i 5539 . . . . . . 7  |-  ( ( i  /  j )  e.  NN  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8478, 83syl 17 . . . . . 6  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8584oveq2d 5808 . . . . 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 12141 . . . 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 4076 . . 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 12125 . . 3  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  e.  _V
8975, 87, 88fvmpt3i 5539 . 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 2299 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 6    /\ wa 360    = wceq 1619    e. wcel 1621   {crab 2522    C_ wss 3127   class class class wbr 3997    e. cmpt 4051   ` cfv 4673  (class class class)co 5792   Fincfn 6831   CCcc 8703   RRcr 8704   1c1 8706    + caddc 8708    x. cmul 8710    / cdiv 9391   NNcn 9714   2c2 9763   ...cfz 10748   ^cexp 11070   sum_csu 12123    || cdivides 12493   logclog 19874  Λcvma 20291   mmucmu 20294
This theorem is referenced by:  selberg  20659
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-disj 3968  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-fi 7133  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9933  df-z 9992  df-dec 10092  df-uz 10198  df-q 10284  df-rp 10322  df-xneg 10419  df-xadd 10420  df-xmul 10421  df-ioo 10626  df-ioc 10627  df-ico 10628  df-icc 10629  df-fz 10749  df-fzo 10837  df-fl 10891  df-mod 10940  df-seq 11013  df-exp 11071  df-fac 11255  df-bc 11282  df-hash 11304  df-shft 11527  df-cj 11549  df-re 11550  df-im 11551  df-sqr 11685  df-abs 11686  df-limsup 11910  df-clim 11927  df-rlim 11928  df-sum 12124  df-ef 12311  df-sin 12313  df-cos 12314  df-pi 12316  df-divides 12494  df-gcd 12648  df-prime 12721  df-pc 12852  df-struct 13112  df-ndx 13113  df-slot 13114  df-base 13115  df-sets 13116  df-ress 13117  df-plusg 13183  df-mulr 13184  df-starv 13185  df-sca 13186  df-vsca 13187  df-tset 13189  df-ple 13190  df-ds 13192  df-hom 13194  df-cco 13195  df-rest 13289  df-topn 13290  df-topgen 13306  df-pt 13307  df-prds 13310  df-xrs 13365  df-0g 13366  df-gsum 13367  df-qtop 13372  df-imas 13373  df-xps 13375  df-mre 13450  df-mrc 13451  df-acs 13453  df-mnd 14329  df-submnd 14378  df-mulg 14454  df-cntz 14755  df-cmn 15053  df-xmet 16335  df-met 16336  df-bl 16337  df-mopn 16338  df-cnfld 16340  df-top 16598  df-bases 16600  df-topon 16601  df-topsp 16602  df-cld 16718  df-ntr 16719  df-cls 16720  df-nei 16797  df-lp 16830  df-perf 16831  df-cn 16919  df-cnp 16920  df-haus 17005  df-tx 17219  df-hmeo 17408  df-fbas 17482  df-fg 17483  df-fil 17503  df-fm 17595  df-flim 17596  df-flf 17597  df-xms 17847  df-ms 17848  df-tms 17849  df-cncf 18344  df-limc 19178  df-dv 19179  df-log 19876  df-vma 20297  df-mu 20300
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