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Theorem logsqvma 22675
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
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 fzfid 11778 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
2 sgmss 22328 . . . 4  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
3 ssfi 7521 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
41, 2, 3syl2anc 654 . . 3  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
5 fzfid 11778 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... d )  e. 
Fin )
6 elrabi 3103 . . . . . . 7  |-  ( d  e.  { x  e.  NN  |  x  ||  N }  ->  d  e.  NN )
76adantl 463 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  NN )
8 sgmss 22328 . . . . . 6  |-  ( d  e.  NN  ->  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d ) )
97, 8syl 16 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d ) )
10 ssfi 7521 . . . . 5  |-  ( ( ( 1 ... d
)  e.  Fin  /\  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d
) )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
115, 9, 10syl2anc 654 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
12 elrabi 3103 . . . . . . . . 9  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  e.  NN )
1312ad2antll 721 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  e.  NN )
14 vmacl 22340 . . . . . . . 8  |-  ( u  e.  NN  ->  (Λ `  u )  e.  RR )
1513, 14syl 16 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
(Λ `  u )  e.  RR )
16 breq1 4283 . . . . . . . . . . . 12  |-  ( x  =  u  ->  (
x  ||  d  <->  u  ||  d
) )
1716elrab 3106 . . . . . . . . . . 11  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  <->  ( u  e.  NN  /\  u  ||  d ) )
1817simprbi 461 . . . . . . . . . 10  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  ||  d )
1918ad2antll 721 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  ||  d )
206ad2antrl 720 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
d  e.  NN )
21 nndivdvds 13523 . . . . . . . . . 10  |-  ( ( d  e.  NN  /\  u  e.  NN )  ->  ( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2220, 13, 21syl2anc 654 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2319, 22mpbid 210 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
( d  /  u
)  e.  NN )
24 vmacl 22340 . . . . . . . 8  |-  ( ( d  /  u )  e.  NN  ->  (Λ `  ( d  /  u
) )  e.  RR )
2523, 24syl 16 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
(Λ `  ( d  /  u ) )  e.  RR )
2615, 25remulcld 9401 . . . . . 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 )
2726recnd 9399 . . . . 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 )
2827anassrs 641 . . . 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 )
2911, 28fsumcl 13193 . . 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 )
30 vmacl 22340 . . . . . 6  |-  ( d  e.  NN  ->  (Λ `  d )  e.  RR )
317, 30syl 16 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  d )  e.  RR )
327nnrpd 11013 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  RR+ )
3332relogcld 21956 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  d )  e.  RR )
3431, 33remulcld 9401 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  RR )
3534recnd 9399 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  CC )
364, 29, 35fsumadd 13198 . 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
) ) ) )
37 id 22 . . . . 5  |-  ( N  e.  NN  ->  N  e.  NN )
38 oveq1 6087 . . . . . . 7  |-  ( d  =  ( u  x.  k )  ->  (
d  /  u )  =  ( ( u  x.  k )  /  u ) )
3938fveq2d 5683 . . . . . 6  |-  ( d  =  ( u  x.  k )  ->  (Λ `  ( d  /  u
) )  =  (Λ `  ( ( u  x.  k )  /  u
) ) )
4039oveq2d 6096 . . . . 5  |-  ( d  =  ( u  x.  k )  ->  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  =  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) ) )
4137, 40, 27fsumdvdscom 22409 . . . 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 ) ) ) )
42 ssrab2 3425 . . . . . . . . . . . . 13  |-  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  NN
43 simpr 458 . . . . . . . . . . . . 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 ) } )
4442, 43sseldi 3342 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  NN )
4544nncnd 10325 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  CC )
46 ssrab2 3425 . . . . . . . . . . . . . 14  |-  { x  e.  NN  |  x  ||  N }  C_  NN
47 simpr 458 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  { x  e.  NN  |  x  ||  N }
)
4846, 47sseldi 3342 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  NN )
4948nncnd 10325 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  CC )
5049adantr 462 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  e.  CC )
5148nnne0d 10353 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  =/=  0 )
5251adantr 462 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  =/=  0 )
5345, 50, 52divcan3d 10099 . . . . . . . . . 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 5683 . . . . . . . . 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 13163 . . . . . . . 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 22405 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e. 
{ x  e.  NN  |  x  ||  N }
)
5746, 56sseldi 3342 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e.  NN )
58 vmasum 22439 . . . . . . . . 9  |-  ( ( N  /  u )  e.  NN  ->  sum_ k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) }  (Λ `  k )  =  ( log `  ( N  /  u ) ) )
5957, 58syl 16 . . . . . . . 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 10987 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
6160adantr 462 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  N  e.  RR+ )
6248nnrpd 11013 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  RR+ )
6361, 62relogdivd 21959 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  ( N  /  u ) )  =  ( ( log `  N
)  -  ( log `  u ) ) )
6455, 59, 633eqtrd 2469 . . . . . . 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 6096 . . . . . 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 11778 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... ( N  /  u ) )  e. 
Fin )
67 sgmss 22328 . . . . . . . . 9  |-  ( ( N  /  u )  e.  NN  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  (
1 ... ( N  /  u ) ) )
6857, 67syl 16 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  (
1 ... ( N  /  u ) ) )
69 ssfi 7521 . . . . . . . 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 654 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  ( N  /  u
) }  e.  Fin )
7148, 14syl 16 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  RR )
7271recnd 9399 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  CC )
73 vmacl 22340 . . . . . . . . . 10  |-  ( k  e.  NN  ->  (Λ `  k )  e.  RR )
7444, 73syl 16 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  (Λ `  k
)  e.  RR )
7574recnd 9399 . . . . . . . 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 2507 . . . . . . 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 13233 . . . . . 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 21911 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( log `  N )  e.  RR )
7978recnd 9399 . . . . . . . 8  |-  ( N  e.  RR+  ->  ( log `  N )  e.  CC )
8061, 79syl 16 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  N )  e.  CC )
8162relogcld 21956 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  RR )
8281recnd 9399 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  CC )
8372, 80, 82subdid 9787 . . . . . 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 2473 . . . . 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 13163 . . . 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 9393 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  N
) )  e.  CC )
8772, 82mulcld 9393 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  u
) )  e.  CC )
884, 86, 87fsumsub 13237 . . . . 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 16 . . . . . . . 8  |-  ( N  e.  NN  ->  ( log `  N )  e.  CC )
9089sqvald 11988 . . . . . . 7  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  =  ( ( log `  N )  x.  ( log `  N ) ) )
91 vmasum 22439 . . . . . . . 8  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
(Λ `  u )  =  ( log `  N
) )
9291oveq1d 6095 . . . . . . 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 13234 . . . . . . 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 2472 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  N
) )  =  ( ( log `  N
) ^ 2 ) )
95 fveq2 5679 . . . . . . . . 9  |-  ( u  =  d  ->  (Λ `  u )  =  (Λ `  d ) )
96 fveq2 5679 . . . . . . . . 9  |-  ( u  =  d  ->  ( log `  u )  =  ( log `  d
) )
9795, 96oveq12d 6098 . . . . . . . 8  |-  ( u  =  d  ->  (
(Λ `  u )  x.  ( log `  u
) )  =  ( (Λ `  d )  x.  ( log `  d
) ) )
9897cbvsumv 13156 . . . . . . 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 11 . . . . . 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 6098 . . . . 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 2465 . . . 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
) ) ) )
10241, 85, 1013eqtrd 2469 . . 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 6095 . 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 11989 . . 3  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  e.  CC )
1054, 35fsumcl 13193 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) )  e.  CC )
106104, 105npcand 9710 . 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 ) )
10736, 103, 1063eqtrd 2469 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    =/= wne 2596   {crab 2709    C_ wss 3316   class class class wbr 4280   ` cfv 5406  (class class class)co 6080   Fincfn 7298   CCcc 9267   RRcr 9268   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274    - cmin 9582    / cdiv 9980   NNcn 10309   2c2 10358   RR+crp 10978   ...cfz 11423   ^cexp 11848   sum_csu 13146    || cdivides 13517   logclog 21890  Λcvma 22313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-pre-sup 9347  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-rp 10979  df-xneg 11076  df-xadd 11077  df-xmul 11078  df-ioo 11291  df-ioc 11292  df-ico 11293  df-icc 11294  df-fz 11424  df-fzo 11532  df-fl 11625  df-mod 11692  df-seq 11790  df-exp 11849  df-fac 12035  df-bc 12062  df-hash 12087  df-shft 12539  df-cj 12571  df-re 12572  df-im 12573  df-sqr 12707  df-abs 12708  df-limsup 12932  df-clim 12949  df-rlim 12950  df-sum 13147  df-ef 13335  df-sin 13337  df-cos 13338  df-pi 13340  df-dvds 13518  df-gcd 13673  df-prm 13746  df-pc 13886  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-sca 14236  df-vsca 14237  df-ip 14238  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-hom 14244  df-cco 14245  df-rest 14343  df-topn 14344  df-0g 14362  df-gsum 14363  df-topgen 14364  df-pt 14365  df-prds 14368  df-xrs 14422  df-qtop 14427  df-imas 14428  df-xps 14430  df-mre 14506  df-mrc 14507  df-acs 14509  df-mnd 15397  df-submnd 15447  df-mulg 15527  df-cntz 15814  df-cmn 16258  df-psmet 17652  df-xmet 17653  df-met 17654  df-bl 17655  df-mopn 17656  df-fbas 17657  df-fg 17658  df-cnfld 17662  df-top 18344  df-bases 18346  df-topon 18347  df-topsp 18348  df-cld 18464  df-ntr 18465  df-cls 18466  df-nei 18543  df-lp 18581  df-perf 18582  df-cn 18672  df-cnp 18673  df-haus 18760  df-tx 18976  df-hmeo 19169  df-fil 19260  df-fm 19352  df-flim 19353  df-flf 19354  df-xms 19736  df-ms 19737  df-tms 19738  df-cncf 20295  df-limc 21182  df-dv 21183  df-log 21892  df-vma 22319
This theorem is referenced by:  logsqvma2  22676
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