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Theorem fsumvma2 20469
Description: Apply fsumvma 20468 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 11051 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
3 elfznn 10835 . . . 4  |-  ( x  e.  ( 1 ... ( |_ `  A
) )  ->  x  e.  NN )
43ssriv 3197 . . 3  |-  ( 1 ... ( |_ `  A ) )  C_  NN
54a1i 10 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) ) 
C_  NN )
6 fsumvma2.2 . . 3  |-  ( ph  ->  A  e.  RR )
7 ppifi 20359 . . 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 3371 . . . . . 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 10835 . . . . 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 12777 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
1615ad2antrl 708 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN )
17 nnnn0 9988 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
1817ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN0 )
1916, 18nnexpcld 11282 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  NN )
2019nnzd 10132 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ZZ )
21 flge 10953 . . . . . 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 10405 . . . . . . . . . . . 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 10132 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  ZZ )
31 relogexp 19965 . . . . . . . . . . . 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 4049 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
3429nnred 9777 . . . . . . . . . . 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 8854 . . . . . . . . . . . . . . 15  |-  0  e.  RR
3736a1i 10 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  e.  RR )
3816nnred 9777 . . . . . . . . . . . . . . 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 9805 . . . . . . . . . . . . . 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 9988 . . . . . . . . . . . . . . . . . 18  |-  ( p  e.  NN  ->  p  e.  NN0 )
4316, 42syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN0 )
4443nn0ge0d 10037 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  <_  p )
45 elicc2 10731 . . . . . . . . . . . . . . . . . 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 8992 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  A
)
5235, 51elrpd 10404 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR+ )
5352relogcld 19990 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  A
)  e.  RR )
54 prmuz2 12792 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
55 eluzelre 10255 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
56 eluz2b2 10306 . . . . . . . . . . . . . 14  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
5756simprbi 450 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
5855, 57rplogcld 19996 . . . . . . . . . . . 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 10451 . . . . . . . . . 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 10433 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
62 flge 10953 . . . . . . . . . . 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 10405 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  RR+ )
6766, 52logled 19994 . . . . . . . . 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 10279 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
7068, 69syl6eleq 2386 . . . . . . . . . . 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 10946 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
73 elfz5 10806 . . . . . . . . . 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 9778 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  CC )
7978exp1d 11256 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  =  p )
8016nnge1d 9804 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  1  <_  p )
8138, 80, 70leexp2ad 11293 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
8279, 81eqbrtrrd 4061 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  <_  ( p ^ k
) )
8319nnred 9777 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  RR )
84 letr 8930 . . . . . . . . . 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 2386 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ( ZZ>= `  1
) )
9114flcld 10946 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  ( |_ `  A )  e.  ZZ )
92 elfz5 10806 . . . . . 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 20468 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   RR+crp 10370   [,]cicc 10675   ...cfz 10798   |_cfl 10940   ^cexp 11120   sum_csu 12174   Primecprime 12774   logclog 19928  Λcvma 20345
This theorem is referenced by:  chpval2  20473  rplogsumlem2  20650  rpvmasumlem  20652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-vma 20351
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