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Theorem logsqvma2 20708
Description: The Möbius inverse of logsqvma 20707. 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 11051 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (
1 ... k )  e. 
Fin )
2 sgmss 20360 . . . . . . . . . 10  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k ) )
3 ssfi 7099 . . . . . . . . . 10  |-  ( ( ( 1 ... k
)  e.  Fin  /\  { x  e.  NN  |  x  ||  k }  C_  ( 1 ... k
) )  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
41, 2, 3syl2anc 642 . . . . . . . . 9  |-  ( k  e.  NN  ->  { x  e.  NN  |  x  ||  k }  e.  Fin )
5 ssrab2 3271 . . . . . . . . . . . 12  |-  { x  e.  NN  |  x  ||  k }  C_  NN
6 simpr 447 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  { x  e.  NN  |  x  ||  k } )
75, 6sseldi 3191 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  d  e.  NN )
8 vmacl 20372 . . . . . . . . . . 11  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
97, 8syl 15 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  d )  e.  RR )
10 dvdsdivcl 20437 . . . . . . . . . . . 12  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  { x  e.  NN  |  x  ||  k } )
115, 10sseldi 3191 . . . . . . . . . . 11  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
k  /  d )  e.  NN )
12 vmacl 20372 . . . . . . . . . . 11  |-  ( ( k  /  d )  e.  NN  ->  (Λ `  ( k  /  d
) )  e.  RR )
1311, 12syl 15 . . . . . . . . . 10  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (Λ `  ( k  /  d
) )  e.  RR )
149, 13remulcld 8879 . . . . . . . . 9  |-  ( ( k  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  e.  RR )
154, 14fsumrecl 12223 . . . . . . . 8  |-  ( k  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  k }  ( (Λ `  d
)  x.  (Λ `  (
k  /  d ) ) )  e.  RR )
16 vmacl 20372 . . . . . . . . 9  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
17 nnrp 10379 . . . . . . . . . 10  |-  ( k  e.  NN  ->  k  e.  RR+ )
1817relogcld 19990 . . . . . . . . 9  |-  ( k  e.  NN  ->  ( log `  k )  e.  RR )
1916, 18remulcld 8879 . . . . . . . 8  |-  ( k  e.  NN  ->  (
(Λ `  k )  x.  ( log `  k
) )  e.  RR )
2015, 19readdcld 8878 . . . . . . 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 8877 . . . . . 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 452 . . . . 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 2296 . . . . 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 5700 . . . 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 3271 . . . . . . . . 9  |-  { x  e.  NN  |  x  ||  n }  C_  NN
26 simpr 447 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  {
x  e.  NN  |  x  ||  n } )
2725, 26sseldi 3191 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  n  e.  NN )  /\  m  e.  {
x  e.  NN  |  x  ||  n } )  ->  m  e.  NN )
28 breq2 4043 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
x  ||  k  <->  x  ||  m
) )
2928rabbidv 2793 . . . . . . . . . . 11  |-  ( k  =  m  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  m } )
30 oveq1 5881 . . . . . . . . . . . . . 14  |-  ( k  =  m  ->  (
k  /  d )  =  ( m  / 
d ) )
3130fveq2d 5545 . . . . . . . . . . . . 13  |-  ( k  =  m  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( m  /  d
) ) )
3231oveq2d 5890 . . . . . . . . . . . 12  |-  ( k  =  m  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3332adantr 451 . . . . . . . . . . 11  |-  ( ( k  =  m  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( m  /  d
) ) ) )
3429, 33sumeq12dv 12195 . . . . . . . . . 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 5541 . . . . . . . . . . 11  |-  ( k  =  m  ->  (Λ `  k )  =  (Λ `  m ) )
36 fveq2 5541 . . . . . . . . . . 11  |-  ( k  =  m  ->  ( log `  k )  =  ( log `  m
) )
3735, 36oveq12d 5892 . . . . . . . . . 10  |-  ( k  =  m  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  m )  x.  ( log `  m
) ) )
3834, 37oveq12d 5892 . . . . . . . . 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 5899 . . . . . . . . 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 5621 . . . . . . . 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 15 . . . . . . 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 12192 . . . . . 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 20707 . . . . . . 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 452 . . . . . 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 2329 . . . . 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 4122 . . . 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 20449 . . 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 5543 . 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 4043 . . . . . 6  |-  ( k  =  N  ->  (
x  ||  k  <->  x  ||  N
) )
5049rabbidv 2793 . . . . 5  |-  ( k  =  N  ->  { x  e.  NN  |  x  ||  k }  =  {
x  e.  NN  |  x  ||  N } )
51 oveq1 5881 . . . . . . . 8  |-  ( k  =  N  ->  (
k  /  d )  =  ( N  / 
d ) )
5251fveq2d 5545 . . . . . . 7  |-  ( k  =  N  ->  (Λ `  ( k  /  d
) )  =  (Λ `  ( N  /  d
) ) )
5352oveq2d 5890 . . . . . 6  |-  ( k  =  N  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5453adantr 451 . . . . 5  |-  ( ( k  =  N  /\  d  e.  { x  e.  NN  |  x  ||  k } )  ->  (
(Λ `  d )  x.  (Λ `  ( k  /  d ) ) )  =  ( (Λ `  d )  x.  (Λ `  ( N  /  d
) ) ) )
5550, 54sumeq12dv 12195 . . . 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 5541 . . . . 5  |-  ( k  =  N  ->  (Λ `  k )  =  (Λ `  N ) )
57 fveq2 5541 . . . . 5  |-  ( k  =  N  ->  ( log `  k )  =  ( log `  N
) )
5856, 57oveq12d 5892 . . . 4  |-  ( k  =  N  ->  (
(Λ `  k )  x.  ( log `  k
) )  =  ( (Λ `  N )  x.  ( log `  N
) ) )
5955, 58oveq12d 5892 . . 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 5621 . 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 5541 . . . . . 6  |-  ( j  =  d  ->  (
mmu `  j )  =  ( mmu `  d ) )
62 oveq2 5882 . . . . . . . 8  |-  ( j  =  d  ->  (
i  /  j )  =  ( i  / 
d ) )
6362fveq2d 5545 . . . . . . 7  |-  ( j  =  d  ->  ( log `  ( i  / 
j ) )  =  ( log `  (
i  /  d ) ) )
6463oveq1d 5889 . . . . . 6  |-  ( j  =  d  ->  (
( log `  (
i  /  j ) ) ^ 2 )  =  ( ( log `  ( i  /  d
) ) ^ 2 ) )
6561, 64oveq12d 5892 . . . . 5  |-  ( j  =  d  ->  (
( mmu `  j
)  x.  ( ( log `  ( i  /  j ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  (
i  /  d ) ) ^ 2 ) ) )
6665cbvsumv 12185 . . . 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 4043 . . . . . 6  |-  ( i  =  N  ->  (
x  ||  i  <->  x  ||  N
) )
6867rabbidv 2793 . . . . 5  |-  ( i  =  N  ->  { x  e.  NN  |  x  ||  i }  =  {
x  e.  NN  |  x  ||  N } )
69 oveq1 5881 . . . . . . . . 9  |-  ( i  =  N  ->  (
i  /  d )  =  ( N  / 
d ) )
7069fveq2d 5545 . . . . . . . 8  |-  ( i  =  N  ->  ( log `  ( i  / 
d ) )  =  ( log `  ( N  /  d ) ) )
7170oveq1d 5889 . . . . . . 7  |-  ( i  =  N  ->  (
( log `  (
i  /  d ) ) ^ 2 )  =  ( ( log `  ( N  /  d
) ) ^ 2 ) )
7271oveq2d 5890 . . . . . 6  |-  ( i  =  N  ->  (
( mmu `  d
)  x.  ( ( log `  ( i  /  d ) ) ^ 2 ) )  =  ( ( mmu `  d )  x.  (
( log `  ( N  /  d ) ) ^ 2 ) ) )
7372adantr 451 . . . . 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 12195 . . . 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 2340 . . 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 3271 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  i }  C_  NN
77 dvdsdivcl 20437 . . . . . . . 8  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  { x  e.  NN  |  x  ||  i } )
7876, 77sseldi 3191 . . . . . . 7  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
i  /  j )  e.  NN )
79 fveq2 5541 . . . . . . . . 9  |-  ( n  =  ( i  / 
j )  ->  ( log `  n )  =  ( log `  (
i  /  j ) ) )
8079oveq1d 5889 . . . . . . . 8  |-  ( n  =  ( i  / 
j )  ->  (
( log `  n
) ^ 2 )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
81 eqid 2296 . . . . . . . 8  |-  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )  =  ( n  e.  NN  |->  ( ( log `  n ) ^ 2 ) )
82 ovex 5899 . . . . . . . 8  |-  ( ( log `  n ) ^ 2 )  e. 
_V
8380, 81, 82fvmpt3i 5621 . . . . . . 7  |-  ( ( i  /  j )  e.  NN  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8478, 83syl 15 . . . . . 6  |-  ( ( i  e.  NN  /\  j  e.  { x  e.  NN  |  x  ||  i } )  ->  (
( n  e.  NN  |->  ( ( log `  n
) ^ 2 ) ) `  ( i  /  j ) )  =  ( ( log `  ( i  /  j
) ) ^ 2 ) )
8584oveq2d 5890 . . . . 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 12192 . . . 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 4118 . . 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 12176 . . 3  |-  sum_ j  e.  { x  e.  NN  |  x  ||  i }  ( ( mmu `  j )  x.  (
( log `  (
i  /  j ) ) ^ 2 ) )  e.  _V
8975, 87, 88fvmpt3i 5621 . 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 2337 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 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    / cdiv 9439   NNcn 9762   2c2 9811   ...cfz 10798   ^cexp 11120   sum_csu 12174    || cdivides 12547   logclog 19928  Λcvma 20345   mmucmu 20348
This theorem is referenced by:  selberg  20713
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-vma 20351  df-mu 20354
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