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Theorem logsqvma 23925
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 12065 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
2 sgmss 23578 . . . 4  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N ) )
3 ssfi 7733 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { x  e.  NN  |  x  ||  N }  C_  ( 1 ... N
) )  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
41, 2, 3syl2anc 659 . . 3  |-  ( N  e.  NN  ->  { x  e.  NN  |  x  ||  N }  e.  Fin )
5 fzfid 12065 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... d )  e. 
Fin )
6 elrabi 3251 . . . . . . 7  |-  ( d  e.  { x  e.  NN  |  x  ||  N }  ->  d  e.  NN )
76adantl 464 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  NN )
8 sgmss 23578 . . . . . 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 7733 . . . . 5  |-  ( ( ( 1 ... d
)  e.  Fin  /\  { x  e.  NN  |  x  ||  d }  C_  ( 1 ... d
) )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
115, 9, 10syl2anc 659 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  { x  e.  NN  |  x  ||  d }  e.  Fin )
12 elrabi 3251 . . . . . . . . 9  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  e.  NN )
1312ad2antll 726 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  e.  NN )
14 vmacl 23590 . . . . . . . 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 4442 . . . . . . . . . . . 12  |-  ( x  =  u  ->  (
x  ||  d  <->  u  ||  d
) )
1716elrab 3254 . . . . . . . . . . 11  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  <->  ( u  e.  NN  /\  u  ||  d ) )
1817simprbi 462 . . . . . . . . . 10  |-  ( u  e.  { x  e.  NN  |  x  ||  d }  ->  u  ||  d )
1918ad2antll 726 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  ->  u  ||  d )
206ad2antrl 725 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( d  e.  {
x  e.  NN  |  x  ||  N }  /\  u  e.  { x  e.  NN  |  x  ||  d } ) )  -> 
d  e.  NN )
21 nndivdvds 14076 . . . . . . . . . 10  |-  ( ( d  e.  NN  /\  u  e.  NN )  ->  ( u  ||  d  <->  ( d  /  u )  e.  NN ) )
2220, 13, 21syl2anc 659 . . . . . . . . 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 23590 . . . . . . . 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 9613 . . . . . 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 9611 . . . . 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 646 . . . 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 13637 . . 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 23590 . . . . . 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 11257 . . . . . 6  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  d  e.  RR+ )
3332relogcld 23176 . . . . 5  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  d )  e.  RR )
3431, 33remulcld 9613 . . . 4  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  RR )
3534recnd 9611 . . 3  |-  ( ( N  e.  NN  /\  d  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  d )  x.  ( log `  d
) )  e.  CC )
364, 29, 35fsumadd 13643 . 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 6277 . . . . . . 7  |-  ( d  =  ( u  x.  k )  ->  (
d  /  u )  =  ( ( u  x.  k )  /  u ) )
3938fveq2d 5852 . . . . . 6  |-  ( d  =  ( u  x.  k )  ->  (Λ `  ( d  /  u
) )  =  (Λ `  ( ( u  x.  k )  /  u
) ) )
4039oveq2d 6286 . . . . 5  |-  ( d  =  ( u  x.  k )  ->  (
(Λ `  u )  x.  (Λ `  ( d  /  u ) ) )  =  ( (Λ `  u
)  x.  (Λ `  (
( u  x.  k
)  /  u ) ) ) )
4137, 40, 27fsumdvdscom 23659 . . . 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 3571 . . . . . . . . . . . . 13  |-  { x  e.  NN  |  x  ||  ( N  /  u
) }  C_  NN
43 simpr 459 . . . . . . . . . . . . 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 3487 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  NN )
4544nncnd 10547 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  k  e.  CC )
46 ssrab2 3571 . . . . . . . . . . . . . 14  |-  { x  e.  NN  |  x  ||  N }  C_  NN
47 simpr 459 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  { x  e.  NN  |  x  ||  N }
)
4846, 47sseldi 3487 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  NN )
4948nncnd 10547 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  CC )
5049adantr 463 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  e.  CC )
5148nnne0d 10576 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  =/=  0 )
5251adantr 463 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  /\  k  e.  { x  e.  NN  |  x  ||  ( N  /  u ) } )  ->  u  =/=  0 )
5345, 50, 52divcan3d 10321 . . . . . . . . . 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 5852 . . . . . . . . 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 13607 . . . . . . . 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 23655 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e. 
{ x  e.  NN  |  x  ||  N }
)
5746, 56sseldi 3487 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( N  /  u )  e.  NN )
58 vmasum 23689 . . . . . . . . 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 11230 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR+ )
6160adantr 463 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  N  e.  RR+ )
6248nnrpd 11257 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  u  e.  RR+ )
6361, 62relogdivd 23179 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  ( N  /  u ) )  =  ( ( log `  N
)  -  ( log `  u ) ) )
6455, 59, 633eqtrd 2499 . . . . . . 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 6286 . . . . . 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 12065 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
1 ... ( N  /  u ) )  e. 
Fin )
67 sgmss 23578 . . . . . . . . 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 7733 . . . . . . . 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 659 . . . . . . 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 9611 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (Λ `  u )  e.  CC )
73 vmacl 23590 . . . . . . . . . 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 9611 . . . . . . . 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 2542 . . . . . . 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 13681 . . . . . 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 23129 . . . . . . . . 9  |-  ( N  e.  RR+  ->  ( log `  N )  e.  RR )
7978recnd 9611 . . . . . . . 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 23176 . . . . . . . 8  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  RR )
8281recnd 9611 . . . . . . 7  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  ( log `  u )  e.  CC )
8372, 80, 82subdid 10008 . . . . . 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 2503 . . . . 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 13607 . . . 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 9605 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  N
) )  e.  CC )
8772, 82mulcld 9605 . . . . . 6  |-  ( ( N  e.  NN  /\  u  e.  { x  e.  NN  |  x  ||  N } )  ->  (
(Λ `  u )  x.  ( log `  u
) )  e.  CC )
884, 86, 87fsumsub 13685 . . . . 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 12289 . . . . . . 7  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  =  ( ( log `  N )  x.  ( log `  N ) ) )
91 vmasum 23689 . . . . . . . 8  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
(Λ `  u )  =  ( log `  N
) )
9291oveq1d 6285 . . . . . . 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 13682 . . . . . . 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 2502 . . . . . 6  |-  ( N  e.  NN  ->  sum_ u  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  u )  x.  ( log `  N
) )  =  ( ( log `  N
) ^ 2 ) )
95 fveq2 5848 . . . . . . . . 9  |-  ( u  =  d  ->  (Λ `  u )  =  (Λ `  d ) )
96 fveq2 5848 . . . . . . . . 9  |-  ( u  =  d  ->  ( log `  u )  =  ( log `  d
) )
9795, 96oveq12d 6288 . . . . . . . 8  |-  ( u  =  d  ->  (
(Λ `  u )  x.  ( log `  u
) )  =  ( (Λ `  d )  x.  ( log `  d
) ) )
9897cbvsumv 13600 . . . . . . 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 6288 . . . . 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 2495 . . . 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 2499 . . 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 6285 . 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 12290 . . 3  |-  ( N  e.  NN  ->  (
( log `  N
) ^ 2 )  e.  CC )
1054, 35fsumcl 13637 . . 3  |-  ( N  e.  NN  ->  sum_ d  e.  { x  e.  NN  |  x  ||  N } 
( (Λ `  d )  x.  ( log `  d
) )  e.  CC )
106104, 105npcand 9926 . 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 2499 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796    / cdiv 10202   NNcn 10531   2c2 10581   RR+crp 11221   ...cfz 11675   ^cexp 12148   sum_csu 13590    || cdvds 14070   logclog 23108  Λcvma 23563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12982  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-limsup 13376  df-clim 13393  df-rlim 13394  df-sum 13591  df-ef 13885  df-sin 13887  df-cos 13888  df-pi 13890  df-dvds 14071  df-gcd 14229  df-prm 14302  df-pc 14445  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437  df-log 23110  df-vma 23569
This theorem is referenced by:  logsqvma2  23926
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