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Theorem logsqvma 20685
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
Dummy variable  k is distinct from all other variables.

Proof of Theorem logsqvma
StepHypRef Expression
1 fzfid 11029 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
2 sgmss 20338 . . . 4  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
3 ssfi 7078 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
41, 2, 3syl2anc 644 . . 3  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
5 fzfid 11029 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... d )  e. 
Fin )
6 ssrab2 3259 . . . . . . . 8  |-  { x  e.  NN  |  x  ||  N }  C_  NN
76sseli 3177 . . . . . . 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 20338 . . . . . 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 7078 . . . . 5  |-  ( ( ( 1 ... d
)  e.  Fin  /\  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d
) )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
125, 10, 11syl2anc 644 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
13 breq1 4027 . . . . . . . . . . 11  |-  ( x  =  u  ->  (
x  ||  d  <->  u  ||  d
) )
1413elrab 2924 . . . . . . . . . 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 711 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  e.  NN )
17 vmacl 20350 . . . . . . . 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 711 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  ||  d )
217ad2antrl 710 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
d  e.  NN )
22 nndivdvds 12531 . . . . . . . . . 10  |-  ( ( d  e.  NN  /\  u  e.  NN )  ->  ( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2321, 16, 22syl2anc 644 . . . . . . . . 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 20350 . . . . . . . 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 8858 . . . . . 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 8856 . . . . 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 631 . . . 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 12200 . . 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 20350 . . . . . 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 10384 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  RR+ )
3433relogcld 19968 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  d )  e.  RR )
3532, 34remulcld 8858 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  RR )
3635recnd 8856 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  CC )
374, 30, 36fsumadd 12205 . 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 5826 . . . . . . 7  |-  ( d  =  ( u  x.  k )  ->  (
d  /  u )  =  ( ( u  x.  k )  /  u ) )
4039fveq2d 5489 . . . . . 6  |-  ( d  =  ( u  x.  k )  ->  (Λ `  ( d  /  u
) )  =  (Λ `  ( ( u  x.  k )  /  u
) ) )
4140oveq2d 5835 . . . . 5  |-  ( d  =  ( u  x.  k )  ->  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  =  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) ) )
4238, 41, 28fsumdvdscom 20419 . . . 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 3259 . . . . . . . . . . . . 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 3179 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  NN )
4645nncnd 9757 . . . . . . . . . . 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 3179 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  NN )
4948nncnd 9757 . . . . . . . . . . . 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 9785 . . . . . . . . . . . 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 9536 . . . . . . . . . 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 5489 . . . . . . . . 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 12170 . . . . . . . 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 20415 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e. 
{ x  e.  NN  |  x  ||  N }
)
576, 56sseldi 3179 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e.  NN )
58 vmasum 20449 . . . . . . . . 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 10358 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
6160adantr 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  N  e.  RR+ )
6248nnrpd 10384 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  RR+ )
6361, 62relogdivd 19971 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  ( N  /  u ) )  =  ( ( log `  N
)  -  ( log `  u ) ) )
6455, 59, 633eqtrd 2320 . . . . . . 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 5835 . . . . . 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 11029 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... ( N  /  u ) )  e. 
Fin )
67 sgmss 20338 . . . . . . . . 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 7078 . . . . . . . 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 644 . . . . . . 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 8856 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  CC )
73 vmacl 20350 . . . . . . . . . 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 8856 . . . . . . . 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 2358 . . . . . . 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 12240 . . . . . 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 19926 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( log `  N )  e.  RR )
7978recnd 8856 . . . . . . . 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 19968 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  RR )
8281recnd 8856 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  CC )
8372, 80, 82subdid 9230 . . . . . 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 2324 . . . . 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 12170 . . . 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 8850 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  N
) )  e.  CC )
8772, 82mulcld 8850 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  u
) )  e.  CC )
884, 86, 87fsumsub 12244 . . . . 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 11236 . . . . . . 7  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  =  ( ( log `  N )  x.  ( log `  N ) ) )
91 vmasum 20449 . . . . . . . 8  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
(Λ `  u )  =  ( log `  N
) )
9291oveq1d 5834 . . . . . . 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 12241 . . . . . . 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 2323 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  N
) )  =  ( ( log `  N
) ^ 2 ) )
95 fveq2 5485 . . . . . . . . 9  |-  ( u  =  d  ->  (Λ `  u )  =  (Λ `  d ) )
96 fveq2 5485 . . . . . . . . 9  |-  ( u  =  d  ->  ( log `  u )  =  ( log `  d
) )
9795, 96oveq12d 5837 . . . . . . . 8  |-  ( u  =  d  ->  (
(Λ `  u )  x.  ( log `  u
) )  =  ( (Λ `  d )  x.  ( log `  d
) ) )
9897cbvsumv 12163 . . . . . . 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 5837 . . . . 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 2316 . . . 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 2320 . . 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 5834 . 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 11237 . . 3  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  e.  CC )
1054, 36fsumcl 12200 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) )  e.  CC )
106104, 105npcand 9156 . 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 2320 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 1624    e. wcel 1685    =/= wne 2447   {crab 2548    C_ wss 3153   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Fincfn 6858   CCcc 8730   RRcr 8731   0cc0 8732   1c1 8733    + caddc 8735    x. cmul 8737    - cmin 9032    / cdiv 9418   NNcn 9741   2c2 9790   RR+crp 10349   ...cfz 10776   ^cexp 11098   sum_csu 12152    || cdivides 12525   logclog 19906  Λcvma 20323
This theorem is referenced by:  logsqvma2  20686
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7337  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-addf 8811  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-of 6039  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-1o 6474  df-2o 6475  df-oadd 6478  df-er 6655  df-map 6769  df-pm 6770  df-ixp 6813  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-fi 7160  df-sup 7189  df-oi 7220  df-card 7567  df-cda 7789  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-4 9801  df-5 9802  df-6 9803  df-7 9804  df-8 9805  df-9 9806  df-10 9807  df-n0 9961  df-z 10020  df-dec 10120  df-uz 10226  df-q 10312  df-rp 10350  df-xneg 10447  df-xadd 10448  df-xmul 10449  df-ioo 10654  df-ioc 10655  df-ico 10656  df-icc 10657  df-fz 10777  df-fzo 10865  df-fl 10919  df-mod 10968  df-seq 11041  df-exp 11099  df-fac 11283  df-bc 11310  df-hash 11332  df-shft 11556  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-limsup 11939  df-clim 11956  df-rlim 11957  df-sum 12153  df-ef 12343  df-sin 12345  df-cos 12346  df-pi 12348  df-dvds 12526  df-gcd 12680  df-prm 12753  df-pc 12884  df-struct 13144  df-ndx 13145  df-slot 13146  df-base 13147  df-sets 13148  df-ress 13149  df-plusg 13215  df-mulr 13216  df-starv 13217  df-sca 13218  df-vsca 13219  df-tset 13221  df-ple 13222  df-ds 13224  df-hom 13226  df-cco 13227  df-rest 13321  df-topn 13322  df-topgen 13338  df-pt 13339  df-prds 13342  df-xrs 13397  df-0g 13398  df-gsum 13399  df-qtop 13404  df-imas 13405  df-xps 13407  df-mre 13482  df-mrc 13483  df-acs 13485  df-mnd 14361  df-submnd 14410  df-mulg 14486  df-cntz 14787  df-cmn 15085  df-xmet 16367  df-met 16368  df-bl 16369  df-mopn 16370  df-cnfld 16372  df-top 16630  df-bases 16632  df-topon 16633  df-topsp 16634  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862  df-perf 16863  df-cn 16951  df-cnp 16952  df-haus 17037  df-tx 17251  df-hmeo 17440  df-fbas 17514  df-fg 17515  df-fil 17535  df-fm 17627  df-flim 17628  df-flf 17629  df-xms 17879  df-ms 17880  df-tms 17881  df-cncf 18376  df-limc 19210  df-dv 19211  df-log 19908  df-vma 20329
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