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Theorem fsumvma2 24005
Description: Apply fsumvma 24004 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 12183 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
3 elfznn 11826 . . . 4  |-  ( x  e.  ( 1 ... ( |_ `  A
) )  ->  x  e.  NN )
43ssriv 3474 . . 3  |-  ( 1 ... ( |_ `  A ) )  C_  NN
54a1i 11 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) ) 
C_  NN )
6 fsumvma2.2 . . 3  |-  ( ph  ->  A  e.  RR )
7 ppifi 23895 . . 3  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
86, 7syl 17 . 2  |-  ( ph  ->  ( ( 0 [,] A )  i^i  Prime )  e.  Fin )
9 elin 3655 . . . . . 6  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
109simprbi 465 . . . . 5  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
11 elfznn 11826 . . . . 5  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
1210, 11anim12i 568 . . . 4  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
1312pm4.71ri 637 . . 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 466 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  A  e.  RR )
15 prmnn 14596 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
1615ad2antrl 732 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN )
17 nnnn0 10876 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
1817ad2antll 733 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN0 )
1916, 18nnexpcld 12434 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  NN )
2019nnzd 11039 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ZZ )
21 flge 12038 . . . . . 6  |-  ( ( A  e.  RR  /\  ( p ^ k
)  e.  ZZ )  ->  ( ( p ^ k )  <_  A 
<->  ( p ^ k
)  <_  ( |_ `  A ) ) )
2214, 20, 21syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
23 simplrl 768 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  Prime )
2423, 15syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  NN )
2524nnrpd 11339 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR+ )
26 simplrr 769 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  NN )
2726nnzd 11039 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  ZZ )
28 relogexp 23410 . . . . . . . . . . 11  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
2925, 27, 28syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  (
p ^ k ) )  =  ( k  x.  ( log `  p
) ) )
3029breq1d 4436 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
3126nnred 10624 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  RR )
3214adantr 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR )
33 0red 9643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  e.  RR )
3416nnred 10624 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  RR )
3534adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR )
3624nngt0d 10653 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  p
)
37 0red 9643 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  e.  RR )
38 nnnn0 10876 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  NN  ->  p  e.  NN0 )
3916, 38syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN0 )
4039nn0ge0d 10928 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  <_  p )
41 elicc2 11699 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
42 df-3an 984 . . . . . . . . . . . . . . . . 17  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
4341, 42syl6bb 264 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
4443baibd 917 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
4537, 14, 34, 40, 44syl22anc 1265 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
4645biimpa 486 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  <_  A
)
4733, 35, 32, 36, 46ltletrd 9794 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  A
)
4832, 47elrpd 11338 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR+ )
4948relogcld 23437 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  A
)  e.  RR )
50 prmuz2 14613 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
51 eluzelre 11169 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
52 eluz2b2 11231 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
5352simprbi 465 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
5451, 53rplogcld 23443 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
5523, 50, 543syl 18 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  p
)  e.  RR+ )
5631, 49, 55lemuldivd 11387 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( k  x.  ( log `  p
) )  <_  ( log `  A )  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
5749, 55rerpdivcld 11369 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
58 flge 12038 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
5957, 27, 58syl2anc 665 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  <_ 
( ( log `  A
)  /  ( log `  p ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6030, 56, 593bitrd 282 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6119adantr 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  NN )
6261nnrpd 11339 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  RR+ )
6362, 48logled 23441 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  ( log `  (
p ^ k ) )  <_  ( log `  A ) ) )
64 simprr 764 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN )
65 nnuz 11194 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
6664, 65syl6eleq 2527 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  ( ZZ>= `  1 )
)
6766adantr 466 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  (
ZZ>= `  1 ) )
6857flcld 12031 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
69 elfz5 11790 . . . . . . . . 9  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7067, 68, 69syl2anc 665 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7160, 63, 703bitr4d 288 . . . . . . 7  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )
7271pm5.32da 645 . . . . . 6  |-  ( (
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 ) ) ) ) ) ) )
7316nncnd 10625 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  CC )
7473exp1d 12408 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  =  p )
7516nnge1d 10652 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  1  <_  p )
7634, 75, 66leexp2ad 12445 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
7774, 76eqbrtrrd 4448 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  <_  ( p ^ k
) )
7819nnred 10624 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  RR )
79 letr 9726 . . . . . . . . . 10  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
8034, 78, 14, 79syl3anc 1264 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
8177, 80mpand 679 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
8281, 45sylibrd 237 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  e.  ( 0 [,] A ) ) )
8382pm4.71rd 639 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p  e.  ( 0 [,] A
)  /\  ( p ^ k )  <_  A ) ) )
849rbaib 914 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  p  e.  (
0 [,] A ) ) )
8584ad2antrl 732 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  e.  ( 0 [,] A
) ) )
8685anbi1d 709 . . . . . 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
)  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) ) ) )
8772, 83, 863bitr4rd 289 . . . . 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 ) )
8819, 65syl6eleq 2527 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ( ZZ>= `  1
) )
8914flcld 12031 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  ( |_ `  A )  e.  ZZ )
90 elfz5 11790 . . . . . 6  |-  ( ( ( p ^ k
)  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9188, 89, 90syl2anc 665 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9222, 87, 913bitr4d 288 . . . 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 ) ) ) )
9392pm5.32da 645 . . 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 ) ) ) ) )
9413, 93syl5bb 260 . 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
) ) ) ) )
95 fsumvma2.3 . 2  |-  ( (
ph  /\  x  e.  ( 1 ... ( |_ `  A ) ) )  ->  B  e.  CC )
96 fsumvma2.4 . 2  |-  ( (
ph  /\  ( x  e.  ( 1 ... ( |_ `  A ) )  /\  (Λ `  x
)  =  0 ) )  ->  B  = 
0 )
971, 2, 5, 8, 94, 95, 96fsumvma 24004 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    i^i cin 3441    C_ wss 3442   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    / cdiv 10268   NNcn 10609   2c2 10659   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   RR+crp 11302   [,]cicc 11638   ...cfz 11782   |_cfl 12023   ^cexp 12269   sum_csu 13730   Primecprime 14593   logclog 23369  Λcvma 23881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-pi 14104  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699  df-log 23371  df-vma 23887
This theorem is referenced by:  chpval2  24009  rplogsumlem2  24186  rpvmasumlem  24188
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