MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fsumvma2 Unicode version

Theorem fsumvma2 20453
Description: Apply fsumvma 20452 for the common case of all numbers less than a real number  A. (Contributed by Mario Carneiro, 30-Apr-2016.)
Hypotheses
Ref Expression
fsumvma2.1  |-  ( x  =  ( p ^
k )  ->  B  =  C )
fsumvma2.2  |-  ( ph  ->  A  e.  RR )
fsumvma2.3  |-  ( (
ph  /\  x  e.  ( 1 ... ( |_ `  A ) ) )  ->  B  e.  CC )
fsumvma2.4  |-  ( (
ph  /\  ( x  e.  ( 1 ... ( |_ `  A ) )  /\  (Λ `  x
)  =  0 ) )  ->  B  = 
0 )
Assertion
Ref Expression
fsumvma2  |-  ( ph  -> 
sum_ x  e.  (
1 ... ( |_ `  A ) ) B  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) C )
Distinct variable groups:    k, p, x, A    x, C    ph, k, p, x    B, k, p
Allowed substitution hints:    B( x)    C( k, p)

Proof of Theorem fsumvma2
StepHypRef Expression
1 fsumvma2.1 . 2  |-  ( x  =  ( p ^
k )  ->  B  =  C )
2 fzfid 11035 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
3 elfznn 10819 . . . 4  |-  ( x  e.  ( 1 ... ( |_ `  A
) )  ->  x  e.  NN )
43ssriv 3184 . . 3  |-  ( 1 ... ( |_ `  A ) )  C_  NN
54a1i 10 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) ) 
C_  NN )
6 fsumvma2.2 . . 3  |-  ( ph  ->  A  e.  RR )
7 ppifi 20343 . . 3  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
86, 7syl 15 . 2  |-  ( ph  ->  ( ( 0 [,] A )  i^i  Prime )  e.  Fin )
9 elin 3358 . . . . . 6  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
109simprbi 450 . . . . 5  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
11 elfznn 10819 . . . . 5  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
1210, 11anim12i 549 . . . 4  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
1312pm4.71ri 614 . . 3  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) ) )
146adantr 451 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  A  e.  RR )
15 prmnn 12761 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
1615ad2antrl 708 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN )
17 nnnn0 9972 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
1817ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN0 )
1916, 18nnexpcld 11266 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  NN )
2019nnzd 10116 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ZZ )
21 flge 10937 . . . . . 6  |-  ( ( A  e.  RR  /\  ( p ^ k
)  e.  ZZ )  ->  ( ( p ^ k )  <_  A 
<->  ( p ^ k
)  <_  ( |_ `  A ) ) )
2214, 20, 21syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
239rbaib 873 . . . . . . . . 9  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  p  e.  (
0 [,] A ) ) )
2423ad2antrl 708 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  e.  ( 0 [,] A
) ) )
2524anbi1d 685 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( p  e.  ( 0 [,] A
)  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) ) )
26 simplrl 736 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  Prime )
2726, 15syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  NN )
2827nnrpd 10389 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR+ )
29 simplrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  NN )
3029nnzd 10116 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  ZZ )
31 relogexp 19949 . . . . . . . . . . . 12  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
3228, 30, 31syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  (
p ^ k ) )  =  ( k  x.  ( log `  p
) ) )
3332breq1d 4033 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
3429nnred 9761 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  RR )
3514adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR )
36 0re 8838 . . . . . . . . . . . . . . 15  |-  0  e.  RR
3736a1i 10 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  e.  RR )
3816nnred 9761 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  RR )
3938adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR )
4027nngt0d 9789 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  p
)
4136a1i 10 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  e.  RR )
42 nnnn0 9972 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  NN  ->  p  e.  NN0 )
4316, 42syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN0 )
4443nn0ge0d 10021 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  <_  p )
45 elicc2 10715 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
46 df-3an 936 . . . . . . . . . . . . . . . . . 18  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
4745, 46syl6bb 252 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
4847baibd 875 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
4941, 14, 38, 44, 48syl22anc 1183 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
5049biimpa 470 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  <_  A
)
5137, 39, 35, 40, 50ltletrd 8976 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  A
)
5235, 51elrpd 10388 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR+ )
5352relogcld 19974 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  A
)  e.  RR )
54 prmuz2 12776 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
55 eluzelre 10239 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
56 eluz2b2 10290 . . . . . . . . . . . . . 14  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
5756simprbi 450 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
5855, 57rplogcld 19980 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
5926, 54, 583syl 18 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  p
)  e.  RR+ )
6034, 53, 59lemuldivd 10435 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( k  x.  ( log `  p
) )  <_  ( log `  A )  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
6153, 59rerpdivcld 10417 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
62 flge 10937 . . . . . . . . . . 11  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6361, 30, 62syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  <_ 
( ( log `  A
)  /  ( log `  p ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6433, 60, 633bitrd 270 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6519adantr 451 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  NN )
6665nnrpd 10389 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  RR+ )
6766, 52logled 19978 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  ( log `  (
p ^ k ) )  <_  ( log `  A ) ) )
68 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN )
69 nnuz 10263 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
7068, 69syl6eleq 2373 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  ( ZZ>= `  1 )
)
7170adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  (
ZZ>= `  1 ) )
7261flcld 10930 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
73 elfz5 10790 . . . . . . . . . 10  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7471, 72, 73syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7564, 67, 743bitr4d 276 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )
7675pm5.32da 622 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  e.  ( 0 [,] A )  /\  ( p ^
k )  <_  A
)  <->  ( p  e.  ( 0 [,] A
)  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) ) )
7725, 76bitr4d 247 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( p  e.  ( 0 [,] A
)  /\  ( p ^ k )  <_  A ) ) )
7816nncnd 9762 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  CC )
7978exp1d 11240 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  =  p )
8016nnge1d 9788 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  1  <_  p )
8138, 80, 70leexp2ad 11277 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
8279, 81eqbrtrrd 4045 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  <_  ( p ^ k
) )
8319nnred 9761 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  RR )
84 letr 8914 . . . . . . . . . 10  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
8538, 83, 14, 84syl3anc 1182 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
8682, 85mpand 656 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
8786, 49sylibrd 225 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  e.  ( 0 [,] A ) ) )
8887pm4.71rd 616 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p  e.  ( 0 [,] A
)  /\  ( p ^ k )  <_  A ) ) )
8977, 88bitr4d 247 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( p ^ k )  <_  A ) )
9019, 69syl6eleq 2373 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ( ZZ>= `  1
) )
9114flcld 10930 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  ( |_ `  A )  e.  ZZ )
92 elfz5 10790 . . . . . 6  |-  ( ( ( p ^ k
)  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9390, 91, 92syl2anc 642 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9422, 89, 933bitr4d 276 . . . 4  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  e.  ( ( 0 [,] A
)  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) )  <->  ( p ^ k )  e.  ( 1 ... ( |_ `  A ) ) ) )
9594pm5.32da 622 . . 3  |-  ( ph  ->  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) )  <-> 
( ( p  e. 
Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  ( 1 ... ( |_
`  A ) ) ) ) )
9613, 95syl5bb 248 . 2  |-  ( ph  ->  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_
`  ( ( log `  A )  /  ( log `  p ) ) ) ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  (
p ^ k )  e.  ( 1 ... ( |_ `  A
) ) ) ) )
97 fsumvma2.3 . 2  |-  ( (
ph  /\  x  e.  ( 1 ... ( |_ `  A ) ) )  ->  B  e.  CC )
98 fsumvma2.4 . 2  |-  ( (
ph  /\  ( x  e.  ( 1 ... ( |_ `  A ) )  /\  (Λ `  x
)  =  0 ) )  ->  B  = 
0 )
991, 2, 5, 8, 96, 97, 98fsumvma 20452 1  |-  ( ph  -> 
sum_ x  e.  (
1 ... ( |_ `  A ) ) B  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) sum_ k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   RR+crp 10354   [,]cicc 10659   ...cfz 10782   |_cfl 10924   ^cexp 11104   sum_csu 12158   Primecprime 12758   logclog 19912  Λcvma 20329
This theorem is referenced by:  chpval2  20457  rplogsumlem2  20634  rpvmasumlem  20636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-vma 20335
  Copyright terms: Public domain W3C validator