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Theorem fsumvma2 22558
Description: Apply fsumvma 22557 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 11800 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
3 elfznn 11483 . . . 4  |-  ( x  e.  ( 1 ... ( |_ `  A
) )  ->  x  e.  NN )
43ssriv 3365 . . 3  |-  ( 1 ... ( |_ `  A ) )  C_  NN
54a1i 11 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) ) 
C_  NN )
6 fsumvma2.2 . . 3  |-  ( ph  ->  A  e.  RR )
7 ppifi 22448 . . 3  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
86, 7syl 16 . 2  |-  ( ph  ->  ( ( 0 [,] A )  i^i  Prime )  e.  Fin )
9 elin 3544 . . . . . 6  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
109simprbi 464 . . . . 5  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
11 elfznn 11483 . . . . 5  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
1210, 11anim12i 566 . . . 4  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
1312pm4.71ri 633 . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  A  e.  RR )
15 prmnn 13771 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
1615ad2antrl 727 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN )
17 nnnn0 10591 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
1817ad2antll 728 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN0 )
1916, 18nnexpcld 12034 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  NN )
2019nnzd 10751 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ZZ )
21 flge 11660 . . . . . 6  |-  ( ( A  e.  RR  /\  ( p ^ k
)  e.  ZZ )  ->  ( ( p ^ k )  <_  A 
<->  ( p ^ k
)  <_  ( |_ `  A ) ) )
2214, 20, 21syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
23 simplrl 759 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  Prime )
2423, 15syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  NN )
2524nnrpd 11031 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR+ )
26 simplrr 760 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  NN )
2726nnzd 10751 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  ZZ )
28 relogexp 22049 . . . . . . . . . . 11  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
2925, 27, 28syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  (
p ^ k ) )  =  ( k  x.  ( log `  p
) ) )
3029breq1d 4307 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
3126nnred 10342 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  RR )
3214adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR )
33 0red 9392 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  e.  RR )
3416nnred 10342 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  RR )
3534adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR )
3624nngt0d 10370 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  p
)
37 0red 9392 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  e.  RR )
38 nnnn0 10591 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  NN  ->  p  e.  NN0 )
3916, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN0 )
4039nn0ge0d 10644 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  <_  p )
41 elicc2 11365 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
42 df-3an 967 . . . . . . . . . . . . . . . . 17  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
4341, 42syl6bb 261 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
4443baibd 900 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
4537, 14, 34, 40, 44syl22anc 1219 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
4645biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  <_  A
)
4733, 35, 32, 36, 46ltletrd 9536 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  A
)
4832, 47elrpd 11030 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR+ )
4948relogcld 22077 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  A
)  e.  RR )
50 prmuz2 13786 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
51 eluzelre 10876 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
52 eluz2b2 10932 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
5352simprbi 464 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
5451, 53rplogcld 22083 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
5523, 50, 543syl 20 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  p
)  e.  RR+ )
5631, 49, 55lemuldivd 11077 . . . . . . . . 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 11059 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
58 flge 11660 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
5957, 27, 58syl2anc 661 . . . . . . . . 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 279 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6119adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  NN )
6261nnrpd 11031 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  RR+ )
6362, 48logled 22081 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  ( log `  (
p ^ k ) )  <_  ( log `  A ) ) )
64 simprr 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN )
65 nnuz 10901 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
6664, 65syl6eleq 2533 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  ( ZZ>= `  1 )
)
6766adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  (
ZZ>= `  1 ) )
6857flcld 11653 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
69 elfz5 11450 . . . . . . . . 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 661 . . . . . . . 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 285 . . . . . . 7  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )
7271pm5.32da 641 . . . . . 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 10343 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  CC )
7473exp1d 12008 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  =  p )
7516nnge1d 10369 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  1  <_  p )
7634, 75, 66leexp2ad 12045 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
7774, 76eqbrtrrd 4319 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  <_  ( p ^ k
) )
7819nnred 10342 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  RR )
79 letr 9473 . . . . . . . . . 10  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
8034, 78, 14, 79syl3anc 1218 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
8177, 80mpand 675 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
8281, 45sylibrd 234 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  e.  ( 0 [,] A ) ) )
8382pm4.71rd 635 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p  e.  ( 0 [,] A
)  /\  ( p ^ k )  <_  A ) ) )
849rbaib 898 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  p  e.  (
0 [,] A ) ) )
8584ad2antrl 727 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  e.  ( 0 [,] A
) ) )
8685anbi1d 704 . . . . . 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 286 . . . . 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 2533 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ( ZZ>= `  1
) )
8914flcld 11653 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  ( |_ `  A )  e.  ZZ )
90 elfz5 11450 . . . . . 6  |-  ( ( ( p ^ k
)  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9188, 89, 90syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9222, 87, 913bitr4d 285 . . . 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 641 . . 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 257 . 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 22557 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3332    C_ wss 3333   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Fincfn 7315   CCcc 9285   RRcr 9286   0cc0 9287   1c1 9288    x. cmul 9292    < clt 9423    <_ cle 9424    / cdiv 9998   NNcn 10327   2c2 10376   NN0cn0 10584   ZZcz 10651   ZZ>=cuz 10866   RR+crp 10996   [,]cicc 11308   ...cfz 11442   |_cfl 11645   ^cexp 11870   sum_csu 13168   Primecprime 13768   logclog 22011  Λcvma 22434
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-fi 7666  df-sup 7696  df-oi 7729  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ioo 11309  df-ioc 11310  df-ico 11311  df-icc 11312  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-bc 12084  df-hash 12109  df-shft 12561  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-limsup 12954  df-clim 12971  df-rlim 12972  df-sum 13169  df-ef 13358  df-sin 13360  df-cos 13361  df-pi 13363  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-rest 14366  df-topn 14367  df-0g 14385  df-gsum 14386  df-topgen 14387  df-pt 14388  df-prds 14391  df-xrs 14445  df-qtop 14450  df-imas 14451  df-xps 14453  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-submnd 15470  df-mulg 15553  df-cntz 15840  df-cmn 16284  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-cnfld 17824  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-lp 18745  df-perf 18746  df-cn 18836  df-cnp 18837  df-haus 18924  df-tx 19140  df-hmeo 19333  df-fil 19424  df-fm 19516  df-flim 19517  df-flf 19518  df-xms 19900  df-ms 19901  df-tms 19902  df-cncf 20459  df-limc 21346  df-dv 21347  df-log 22013  df-vma 22440
This theorem is referenced by:  chpval2  22562  rplogsumlem2  22739  rpvmasumlem  22741
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