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Theorem logsqvma 20618
Description: A formula for  log ^
2 ( N ) in terms of the primes. Equation 10.4.6 of [Shapiro], p. 418. (Contributed by Mario Carneiro, 13-May-2016.)
Assertion
Ref Expression
logsqvma  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( sum_ u  e.  {
x  e.  NN  |  x  ||  d }  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  +  ( (Λ `  d
)  x.  ( log `  d ) ) )  =  ( ( log `  N ) ^ 2 ) )
Distinct variable group:    u, d, x, N

Proof of Theorem logsqvma
StepHypRef Expression
1 fzfid 10966 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
2 sgmss 20271 . . . 4  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
3 ssfi 7016 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
41, 2, 3syl2anc 645 . . 3  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
5 fzfid 10966 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... d )  e. 
Fin )
6 ssrab2 3200 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  N }  C_  NN
76sseli 3118 . . . . . . 7  |-  ( d  e.  { x  e.  NN  |  x  ||  N }  ->  d  e.  NN )
87adantl 454 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  NN )
9 sgmss 20271 . . . . . 6  |-  ( d  e.  NN  ->  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d ) )
108, 9syl 17 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d ) )
11 ssfi 7016 . . . . 5  |-  ( ( ( 1 ... d
)  e.  Fin  /\  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d
) )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
125, 10, 11syl2anc 645 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
13 breq1 3966 . . . . . . . . . . 11  |-  ( x  =  u  ->  (
x  ||  d  <->  u  ||  d
) )
1413elrab 2874 . . . . . . . . . 10  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  <->  ( u  e.  NN  /\  u  ||  d ) )
1514simplbi 448 . . . . . . . . 9  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  e.  NN )
1615ad2antll 712 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  e.  NN )
17 vmacl 20283 . . . . . . . 8  |-  ( u  e.  NN  ->  (Λ `  u )  e.  RR )
1816, 17syl 17 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
(Λ `  u )  e.  RR )
1914simprbi 452 . . . . . . . . . 10  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  ||  d )
2019ad2antll 712 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  ||  d )
217ad2antrl 711 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
d  e.  NN )
22 nndivdivides 12464 . . . . . . . . . 10  |-  ( ( d  e.  NN  /\  u  e.  NN )  ->  ( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2321, 16, 22syl2anc 645 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2420, 23mpbid 203 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( d  /  u
)  e.  NN )
25 vmacl 20283 . . . . . . . 8  |-  ( ( d  /  u )  e.  NN  ->  (Λ `  ( d  /  u
) )  e.  RR )
2624, 25syl 17 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
(Λ `  ( d  /  u ) )  e.  RR )
2718, 26remulcld 8796 . . . . . 6  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( (Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  e.  RR )
2827recnd 8794 . . . . 5  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( (Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  e.  CC )
2928anassrs 632 . . . 4  |-  ( ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  /\  u  e.  { x  e.  NN  |  x  ||  d } )  ->  ( (Λ `  u )  x.  (Λ `  ( d  /  u
) ) )  e.  CC )
3012, 29fsumcl 12136 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  sum_ u  e.  { x  e.  NN  |  x  ||  d }  ( (Λ `  u
)  x.  (Λ `  (
d  /  u ) ) )  e.  CC )
31 vmacl 20283 . . . . . 6  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
328, 31syl 17 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  d )  e.  RR )
338nnrpd 10321 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  RR+ )
3433relogcld 19901 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  d )  e.  RR )
3532, 34remulcld 8796 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  RR )
3635recnd 8794 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  CC )
374, 30, 36fsumadd 12141 . 2  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( sum_ u  e.  {
x  e.  NN  |  x  ||  d }  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  +  ( (Λ `  d
)  x.  ( log `  d ) ) )  =  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } sum_ u  e.  { x  e.  NN  |  x  ||  d }  ( (Λ `  u )  x.  (Λ `  ( d  /  u
) ) )  + 
sum_ d  e.  {
x  e.  NN  |  x  ||  N }  (
(Λ `  d )  x.  ( log `  d
) ) ) )
38 id 21 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
39 oveq1 5764 . . . . . . 7  |-  ( d  =  ( u  x.  k )  ->  (
d  /  u )  =  ( ( u  x.  k )  /  u ) )
4039fveq2d 5427 . . . . . 6  |-  ( d  =  ( u  x.  k )  ->  (Λ `  ( d  /  u
) )  =  (Λ `  ( ( u  x.  k )  /  u
) ) )
4140oveq2d 5773 . . . . 5  |-  ( d  =  ( u  x.  k )  ->  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  =  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) ) )
4238, 41, 28fsumdvdscom 20352 . . . 4  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } sum_ u  e.  { x  e.  NN  |  x  ||  d }  ( (Λ `  u )  x.  (Λ `  ( d  /  u
) ) )  = 
sum_ u  e.  { x  e.  NN  |  x  ||  N } sum_ k  e.  {
x  e.  NN  |  x  ||  ( N  /  u ) }  (
(Λ `  u )  x.  (Λ `  ( (
u  x.  k )  /  u ) ) ) )
43 ssrab2 3200 . . . . . . . . . . . . 13  |-  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  NN
44 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )
4543, 44sseldi 3120 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  NN )
4645nncnd 9695 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  CC )
47 simpr 449 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  { x  e.  NN  |  x  ||  N }
)
486, 47sseldi 3120 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  NN )
4948nncnd 9695 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  CC )
5049adantr 453 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  e.  CC )
5148nnne0d 9723 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  =/=  0 )
5251adantr 453 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  =/=  0 )
5346, 50, 52divcan3d 9474 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  ( (
u  x.  k )  /  u )  =  k )
5453fveq2d 5427 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  (Λ `  (
( u  x.  k
)  /  u ) )  =  (Λ `  k
) )
5554sumeq2dv 12106 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  ( (
u  x.  k )  /  u ) )  =  sum_ k  e.  {
x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  k ) )
56 dvdsdivcl 20348 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e. 
{ x  e.  NN  |  x  ||  N }
)
576, 56sseldi 3120 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e.  NN )
58 vmasum 20382 . . . . . . . . 9  |-  ( ( N  /  u )  e.  NN  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  k )  =  ( log `  ( N  /  u ) ) )
5957, 58syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  k )  =  ( log `  ( N  /  u ) ) )
60 nnrp 10295 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
6160adantr 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  N  e.  RR+ )
6248nnrpd 10321 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  RR+ )
6361, 62relogdivd 19904 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  ( N  /  u ) )  =  ( ( log `  N
)  -  ( log `  u ) ) )
6455, 59, 633eqtrd 2292 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  ( (
u  x.  k )  /  u ) )  =  ( ( log `  N )  -  ( log `  u ) ) )
6564oveq2d 5773 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x. 
sum_ k  e.  {
x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  ( ( u  x.  k )  /  u
) ) )  =  ( (Λ `  u
)  x.  ( ( log `  N )  -  ( log `  u
) ) ) )
66 fzfid 10966 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... ( N  /  u ) )  e. 
Fin )
67 sgmss 20271 . . . . . . . . 9  |-  ( ( N  /  u )  e.  NN  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  (
1 ... ( N  /  u ) ) )
6857, 67syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  (
1 ... ( N  /  u ) ) )
69 ssfi 7016 . . . . . . . 8  |-  ( ( ( 1 ... ( N  /  u ) )  e.  Fin  /\  {
x  e.  NN  |  x  ||  ( N  /  u ) }  C_  ( 1 ... ( N  /  u ) ) )  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  e.  Fin )
7066, 68, 69syl2anc 645 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  e.  Fin )
7148, 17syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  RR )
7271recnd 8794 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  CC )
73 vmacl 20283 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
7445, 73syl 17 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  (Λ `  k
)  e.  RR )
7574recnd 8794 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  (Λ `  k
)  e.  CC )
7654, 75eqeltrd 2330 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  (Λ `  (
( u  x.  k
)  /  u ) )  e.  CC )
7770, 72, 76fsummulc2 12176 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x. 
sum_ k  e.  {
x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  ( ( u  x.  k )  /  u
) ) )  = 
sum_ k  e.  {
x  e.  NN  |  x  ||  ( N  /  u ) }  (
(Λ `  u )  x.  (Λ `  ( (
u  x.  k )  /  u ) ) ) )
78 relogcl 19859 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( log `  N )  e.  RR )
7978recnd 8794 . . . . . . . 8  |-  ( N  e.  RR+  ->  ( log `  N )  e.  CC )
8061, 79syl 17 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  N )  e.  CC )
8162relogcld 19901 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  RR )
8281recnd 8794 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  CC )
8372, 80, 82subdid 9168 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( ( log `  N
)  -  ( log `  u ) ) )  =  ( ( (Λ `  u )  x.  ( log `  N ) )  -  ( (Λ `  u
)  x.  ( log `  u ) ) ) )
8465, 77, 833eqtr3d 2296 . . . . 5  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) )  =  ( ( (Λ `  u
)  x.  ( log `  N ) )  -  ( (Λ `  u )  x.  ( log `  u
) ) ) )
8584sumeq2dv 12106 . . . 4  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u
) }  ( (Λ `  u )  x.  (Λ `  ( ( u  x.  k )  /  u
) ) )  = 
sum_ u  e.  { x  e.  NN  |  x  ||  N }  ( (
(Λ `  u )  x.  ( log `  N
) )  -  (
(Λ `  u )  x.  ( log `  u
) ) ) )
8672, 80mulcld 8788 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  N
) )  e.  CC )
8772, 82mulcld 8788 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  u
) )  e.  CC )
884, 86, 87fsumsub 12180 . . . . 5  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( ( (Λ `  u
)  x.  ( log `  N ) )  -  ( (Λ `  u )  x.  ( log `  u
) ) )  =  ( sum_ u  e.  {
x  e.  NN  |  x  ||  N }  (
(Λ `  u )  x.  ( log `  N
) )  -  sum_ u  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  u )  x.  ( log `  u ) ) ) )
8960, 79syl 17 . . . . . . . 8  |-  ( N  e.  NN  ->  ( log `  N )  e.  CC )
9089sqvald 11173 . . . . . . 7  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  =  ( ( log `  N )  x.  ( log `  N ) ) )
91 vmasum 20382 . . . . . . . 8  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
(Λ `  u )  =  ( log `  N
) )
9291oveq1d 5772 . . . . . . 7  |-  ( N  e.  NN  ->  ( sum_ u  e.  { x  e.  NN  |  x  ||  N }  (Λ `  u
)  x.  ( log `  N ) )  =  ( ( log `  N
)  x.  ( log `  N ) ) )
934, 89, 72fsummulc1 12177 . . . . . . 7  |-  ( N  e.  NN  ->  ( sum_ u  e.  { x  e.  NN  |  x  ||  N }  (Λ `  u
)  x.  ( log `  N ) )  = 
sum_ u  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  u )  x.  ( log `  N ) ) )
9490, 92, 933eqtr2rd 2295 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  N
) )  =  ( ( log `  N
) ^ 2 ) )
95 fveq2 5423 . . . . . . . . 9  |-  ( u  =  d  ->  (Λ `  u )  =  (Λ `  d ) )
96 fveq2 5423 . . . . . . . . 9  |-  ( u  =  d  ->  ( log `  u )  =  ( log `  d
) )
9795, 96oveq12d 5775 . . . . . . . 8  |-  ( u  =  d  ->  (
(Λ `  u )  x.  ( log `  u
) )  =  ( (Λ `  d )  x.  ( log `  d
) ) )
9897cbvsumv 12099 . . . . . . 7  |-  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  u
) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  ( log `  d ) )
9998a1i 12 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  u
) )  =  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  ( log `  d ) ) )
10094, 99oveq12d 5775 . . . . 5  |-  ( N  e.  NN  ->  ( sum_ u  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  u )  x.  ( log `  N ) )  -  sum_ u  e.  {
x  e.  NN  |  x  ||  N }  (
(Λ `  u )  x.  ( log `  u
) ) )  =  ( ( ( log `  N ) ^ 2 )  -  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) ) ) )
10188, 100eqtrd 2288 . . . 4  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( ( (Λ `  u
)  x.  ( log `  N ) )  -  ( (Λ `  u )  x.  ( log `  u
) ) )  =  ( ( ( log `  N ) ^ 2 )  -  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) ) ) )
10242, 85, 1013eqtrd 2292 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } sum_ u  e.  { x  e.  NN  |  x  ||  d }  ( (Λ `  u )  x.  (Λ `  ( d  /  u
) ) )  =  ( ( ( log `  N ) ^ 2 )  -  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) ) ) )
103102oveq1d 5772 . 2  |-  ( N  e.  NN  ->  ( sum_ d  e.  { x  e.  NN  |  x  ||  N } sum_ u  e.  {
x  e.  NN  |  x  ||  d }  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  +  sum_ d  e.  {
x  e.  NN  |  x  ||  N }  (
(Λ `  d )  x.  ( log `  d
) ) )  =  ( ( ( ( log `  N ) ^ 2 )  -  sum_ d  e.  { x  e.  NN  |  x  ||  N }  ( (Λ `  d )  x.  ( log `  d ) ) )  +  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) ) ) )
10489sqcld 11174 . . 3  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  e.  CC )
1054, 36fsumcl 12136 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) )  e.  CC )
106104, 105npcand 9094 . 2  |-  ( N  e.  NN  ->  (
( ( ( log `  N ) ^ 2 )  -  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) ) )  + 
sum_ d  e.  {
x  e.  NN  |  x  ||  N }  (
(Λ `  d )  x.  ( log `  d
) ) )  =  ( ( log `  N
) ^ 2 ) )
10737, 103, 1063eqtrd 2292 1  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( sum_ u  e.  {
x  e.  NN  |  x  ||  d }  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  +  ( (Λ `  d
)  x.  ( log `  d ) ) )  =  ( ( log `  N ) ^ 2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   {crab 2519    C_ wss 3094   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Fincfn 6796   CCcc 8668   RRcr 8669   0cc0 8670   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   RR+crp 10286   ...cfz 10713   ^cexp 11035   sum_csu 12088    || cdivides 12458   logclog 19839  Λcvma 20256
This theorem is referenced by:  logsqvma2  20619
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-vma 20262
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