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Theorem logsqvma 22766
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 11787 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
2 sgmss 22419 . . . 4  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
3 ssfi 7525 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
41, 2, 3syl2anc 661 . . 3  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
5 fzfid 11787 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... d )  e. 
Fin )
6 elrabi 3109 . . . . . . 7  |-  ( d  e.  { x  e.  NN  |  x  ||  N }  ->  d  e.  NN )
76adantl 466 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  NN )
8 sgmss 22419 . . . . . 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 7525 . . . . 5  |-  ( ( ( 1 ... d
)  e.  Fin  /\  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d
) )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
115, 9, 10syl2anc 661 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
12 elrabi 3109 . . . . . . . . 9  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  e.  NN )
1312ad2antll 728 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  e.  NN )
14 vmacl 22431 . . . . . . . 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 4290 . . . . . . . . . . . 12  |-  ( x  =  u  ->  (
x  ||  d  <->  u  ||  d
) )
1716elrab 3112 . . . . . . . . . . 11  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  <->  ( u  e.  NN  /\  u  ||  d ) )
1817simprbi 464 . . . . . . . . . 10  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  ||  d )
1918ad2antll 728 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  ||  d )
206ad2antrl 727 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
d  e.  NN )
21 nndivdvds 13533 . . . . . . . . . 10  |-  ( ( d  e.  NN  /\  u  e.  NN )  ->  ( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2220, 13, 21syl2anc 661 . . . . . . . . 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 22431 . . . . . . . 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 9406 . . . . . 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 9404 . . . . 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 648 . . . 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 13202 . . 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 22431 . . . . . 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 11018 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  RR+ )
3332relogcld 22047 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  d )  e.  RR )
3431, 33remulcld 9406 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  RR )
3534recnd 9404 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  CC )
364, 29, 35fsumadd 13207 . 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 6093 . . . . . . 7  |-  ( d  =  ( u  x.  k )  ->  (
d  /  u )  =  ( ( u  x.  k )  /  u ) )
3938fveq2d 5690 . . . . . 6  |-  ( d  =  ( u  x.  k )  ->  (Λ `  ( d  /  u
) )  =  (Λ `  ( ( u  x.  k )  /  u
) ) )
4039oveq2d 6102 . . . . 5  |-  ( d  =  ( u  x.  k )  ->  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  =  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) ) )
4137, 40, 27fsumdvdscom 22500 . . . 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 3432 . . . . . . . . . . . . 13  |-  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  NN
43 simpr 461 . . . . . . . . . . . . 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 3349 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  NN )
4544nncnd 10330 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  CC )
46 ssrab2 3432 . . . . . . . . . . . . . 14  |-  { x  e.  NN  |  x  ||  N }  C_  NN
47 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  { x  e.  NN  |  x  ||  N }
)
4846, 47sseldi 3349 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  NN )
4948nncnd 10330 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  CC )
5049adantr 465 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  e.  CC )
5148nnne0d 10358 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  =/=  0 )
5251adantr 465 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  =/=  0 )
5345, 50, 52divcan3d 10104 . . . . . . . . . 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 5690 . . . . . . . . 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 13172 . . . . . . . 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 22496 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e. 
{ x  e.  NN  |  x  ||  N }
)
5746, 56sseldi 3349 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e.  NN )
58 vmasum 22530 . . . . . . . . 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 10992 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
6160adantr 465 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  N  e.  RR+ )
6248nnrpd 11018 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  RR+ )
6361, 62relogdivd 22050 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  ( N  /  u ) )  =  ( ( log `  N
)  -  ( log `  u ) ) )
6455, 59, 633eqtrd 2474 . . . . . . 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 6102 . . . . . 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 11787 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... ( N  /  u ) )  e. 
Fin )
67 sgmss 22419 . . . . . . . . 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 7525 . . . . . . . 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 661 . . . . . . 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 9404 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  CC )
73 vmacl 22431 . . . . . . . . . 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 9404 . . . . . . . 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 2512 . . . . . . 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 13243 . . . . . 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 22002 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( log `  N )  e.  RR )
7978recnd 9404 . . . . . . . 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 22047 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  RR )
8281recnd 9404 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  CC )
8372, 80, 82subdid 9792 . . . . . 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 2478 . . . . 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 13172 . . . 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 9398 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  N
) )  e.  CC )
8772, 82mulcld 9398 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  u
) )  e.  CC )
884, 86, 87fsumsub 13247 . . . . 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 11997 . . . . . . 7  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  =  ( ( log `  N )  x.  ( log `  N ) ) )
91 vmasum 22530 . . . . . . . 8  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
(Λ `  u )  =  ( log `  N
) )
9291oveq1d 6101 . . . . . . 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 13244 . . . . . . 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 2477 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  N
) )  =  ( ( log `  N
) ^ 2 ) )
95 fveq2 5686 . . . . . . . . 9  |-  ( u  =  d  ->  (Λ `  u )  =  (Λ `  d ) )
96 fveq2 5686 . . . . . . . . 9  |-  ( u  =  d  ->  ( log `  u )  =  ( log `  d
) )
9795, 96oveq12d 6104 . . . . . . . 8  |-  ( u  =  d  ->  (
(Λ `  u )  x.  ( log `  u
) )  =  ( (Λ `  d )  x.  ( log `  d
) ) )
9897cbvsumv 13165 . . . . . . 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 6104 . . . . 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 2470 . . . 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 2474 . . 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 6101 . 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 11998 . . 3  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  e.  CC )
1054, 35fsumcl 13202 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) )  e.  CC )
106104, 105npcand 9715 . 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 2474 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 1369    e. wcel 1756    =/= wne 2601   {crab 2714    C_ wss 3323   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Fincfn 7302   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    - cmin 9587    / cdiv 9985   NNcn 10314   2c2 10363   RR+crp 10983   ...cfz 11429   ^cexp 11857   sum_csu 13155    || cdivides 13527   logclog 21981  Λcvma 22404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ioc 11297  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-shft 12548  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-limsup 12941  df-clim 12958  df-rlim 12959  df-sum 13156  df-ef 13345  df-sin 13347  df-cos 13348  df-pi 13350  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-lp 18715  df-perf 18716  df-cn 18806  df-cnp 18807  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cncf 20429  df-limc 21316  df-dv 21317  df-log 21983  df-vma 22410
This theorem is referenced by:  logsqvma2  22767
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