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Theorem fsumvma2 20999
Description: Apply fsumvma 20998 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 11313 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) )  e.  Fin )
3 elfznn 11081 . . . 4  |-  ( x  e.  ( 1 ... ( |_ `  A
) )  ->  x  e.  NN )
43ssriv 3353 . . 3  |-  ( 1 ... ( |_ `  A ) )  C_  NN
54a1i 11 . 2  |-  ( ph  ->  ( 1 ... ( |_ `  A ) ) 
C_  NN )
6 fsumvma2.2 . . 3  |-  ( ph  ->  A  e.  RR )
7 ppifi 20889 . . 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 3531 . . . . . 6  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  ( p  e.  ( 0 [,] A
)  /\  p  e.  Prime ) )
109simprbi 452 . . . . 5  |-  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  ->  p  e. 
Prime )
11 elfznn 11081 . . . . 5  |-  ( k  e.  ( 1 ... ( |_ `  (
( log `  A
)  /  ( log `  p ) ) ) )  ->  k  e.  NN )
1210, 11anim12i 551 . . . 4  |-  ( ( p  e.  ( ( 0 [,] A )  i^i  Prime )  /\  k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
1312pm4.71ri 616 . . 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 453 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  A  e.  RR )
15 prmnn 13083 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  NN )
1615ad2antrl 710 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN )
17 nnnn0 10229 . . . . . . . . 9  |-  ( k  e.  NN  ->  k  e.  NN0 )
1817ad2antll 711 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN0 )
1916, 18nnexpcld 11545 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  NN )
2019nnzd 10375 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ZZ )
21 flge 11215 . . . . . 6  |-  ( ( A  e.  RR  /\  ( p ^ k
)  e.  ZZ )  ->  ( ( p ^ k )  <_  A 
<->  ( p ^ k
)  <_  ( |_ `  A ) ) )
2214, 20, 21syl2anc 644 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
23 simplrl 738 . . . . . . . . . . . . 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 10648 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR+ )
26 simplrr 739 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  NN )
2726nnzd 10375 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  ZZ )
28 relogexp 20491 . . . . . . . . . . 11  |-  ( ( p  e.  RR+  /\  k  e.  ZZ )  ->  ( log `  ( p ^
k ) )  =  ( k  x.  ( log `  p ) ) )
2925, 27, 28syl2anc 644 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  (
p ^ k ) )  =  ( k  x.  ( log `  p
) ) )
3029breq1d 4223 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  ( k  x.  ( log `  p
) )  <_  ( log `  A ) ) )
3126nnred 10016 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  RR )
3214adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR )
33 0re 9092 . . . . . . . . . . . . . 14  |-  0  e.  RR
3433a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  e.  RR )
3516nnred 10016 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  RR )
3635adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  e.  RR )
3724nngt0d 10044 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  p
)
3833a1i 11 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  e.  RR )
39 nnnn0 10229 . . . . . . . . . . . . . . . . 17  |-  ( p  e.  NN  ->  p  e.  NN0 )
4016, 39syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  NN0 )
4140nn0ge0d 10278 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  0  <_  p )
42 elicc2 10976 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
43 df-3an 939 . . . . . . . . . . . . . . . . 17  |-  ( ( p  e.  RR  /\  0  <_  p  /\  p  <_  A )  <->  ( (
p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) )
4442, 43syl6bb 254 . . . . . . . . . . . . . . . 16  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( ( p  e.  RR  /\  0  <_  p )  /\  p  <_  A ) ) )
4544baibd 877 . . . . . . . . . . . . . . 15  |-  ( ( ( 0  e.  RR  /\  A  e.  RR )  /\  ( p  e.  RR  /\  0  <_  p ) )  -> 
( p  e.  ( 0 [,] A )  <-> 
p  <_  A )
)
4638, 14, 35, 41, 45syl22anc 1186 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( 0 [,] A )  <->  p  <_  A ) )
4746biimpa 472 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  p  <_  A
)
4834, 36, 32, 37, 47ltletrd 9231 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  0  <  A
)
4932, 48elrpd 10647 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  A  e.  RR+ )
5049relogcld 20519 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  A
)  e.  RR )
51 prmuz2 13098 . . . . . . . . . . 11  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
52 eluzelre 10498 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  p  e.  RR )
53 eluz2b2 10549 . . . . . . . . . . . . 13  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
5453simprbi 452 . . . . . . . . . . . 12  |-  ( p  e.  ( ZZ>= `  2
)  ->  1  <  p )
5552, 54rplogcld 20525 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  ->  ( log `  p )  e.  RR+ )
5623, 51, 553syl 19 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( log `  p
)  e.  RR+ )
5731, 50, 56lemuldivd 10694 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( k  x.  ( log `  p
) )  <_  ( log `  A )  <->  k  <_  ( ( log `  A
)  /  ( log `  p ) ) ) )
5850, 56rerpdivcld 10676 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  A )  /  ( log `  p ) )  e.  RR )
59 flge 11215 . . . . . . . . . 10  |-  ( ( ( ( log `  A
)  /  ( log `  p ) )  e.  RR  /\  k  e.  ZZ )  ->  (
k  <_  ( ( log `  A )  / 
( log `  p
) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6058, 27, 59syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  <_ 
( ( log `  A
)  /  ( log `  p ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6130, 57, 603bitrd 272 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( log `  ( p ^ k
) )  <_  ( log `  A )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
6219adantr 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  NN )
6362nnrpd 10648 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( p ^
k )  e.  RR+ )
6463, 49logled 20523 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  ( log `  (
p ^ k ) )  <_  ( log `  A ) ) )
65 simprr 735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  NN )
66 nnuz 10522 . . . . . . . . . . 11  |-  NN  =  ( ZZ>= `  1 )
6765, 66syl6eleq 2527 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  k  e.  ( ZZ>= `  1 )
)
6867adantr 453 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  k  e.  (
ZZ>= `  1 ) )
6958flcld 11208 . . . . . . . . 9  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) )  e.  ZZ )
70 elfz5 11052 . . . . . . . . 9  |-  ( ( k  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  ( ( log `  A )  /  ( log `  p ) ) )  e.  ZZ )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7168, 69, 70syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( k  e.  ( 1 ... ( |_ `  ( ( log `  A )  /  ( log `  p ) ) ) )  <->  k  <_  ( |_ `  ( ( log `  A )  /  ( log `  p
) ) ) ) )
7261, 64, 713bitr4d 278 . . . . . . 7  |-  ( ( ( ph  /\  (
p  e.  Prime  /\  k  e.  NN ) )  /\  p  e.  ( 0 [,] A ) )  ->  ( ( p ^ k )  <_  A 
<->  k  e.  ( 1 ... ( |_ `  ( ( log `  A
)  /  ( log `  p ) ) ) ) ) )
7372pm5.32da 624 . . . . . 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 ) ) ) ) ) ) )
7416nncnd 10017 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  e.  CC )
7574exp1d 11519 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  =  p )
7616nnge1d 10043 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  1  <_  p )
7735, 76, 67leexp2ad 11556 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ 1 )  <_  ( p ^
k ) )
7875, 77eqbrtrrd 4235 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  p  <_  ( p ^ k
) )
7919nnred 10016 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  RR )
80 letr 9168 . . . . . . . . . 10  |-  ( ( p  e.  RR  /\  ( p ^ k
)  e.  RR  /\  A  e.  RR )  ->  ( ( p  <_ 
( p ^ k
)  /\  ( p ^ k )  <_  A )  ->  p  <_  A ) )
8135, 79, 14, 80syl3anc 1185 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p  <_  (
p ^ k )  /\  ( p ^
k )  <_  A
)  ->  p  <_  A ) )
8278, 81mpand 658 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  <_  A ) )
8382, 46sylibrd 227 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  ->  p  e.  ( 0 [,] A ) ) )
8483pm4.71rd 618 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  <_  A  <->  ( p  e.  ( 0 [,] A
)  /\  ( p ^ k )  <_  A ) ) )
859rbaib 875 . . . . . . . 8  |-  ( p  e.  Prime  ->  ( p  e.  ( ( 0 [,] A )  i^i 
Prime )  <->  p  e.  (
0 [,] A ) ) )
8685ad2antrl 710 . . . . . . 7  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p  e.  ( ( 0 [,] A )  i^i  Prime )  <->  p  e.  ( 0 [,] A
) ) )
8786anbi1d 687 . . . . . 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 ) ) ) ) ) ) )
8873, 84, 873bitr4rd 279 . . . . 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 ) )
8919, 66syl6eleq 2527 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
p ^ k )  e.  ( ZZ>= `  1
) )
9014flcld 11208 . . . . . 6  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  ( |_ `  A )  e.  ZZ )
91 elfz5 11052 . . . . . 6  |-  ( ( ( p ^ k
)  e.  ( ZZ>= ` 
1 )  /\  ( |_ `  A )  e.  ZZ )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9289, 90, 91syl2anc 644 . . . . 5  |-  ( (
ph  /\  ( p  e.  Prime  /\  k  e.  NN ) )  ->  (
( p ^ k
)  e.  ( 1 ... ( |_ `  A ) )  <->  ( p ^ k )  <_ 
( |_ `  A
) ) )
9322, 88, 923bitr4d 278 . . . 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 ) ) ) )
9493pm5.32da 624 . . 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 ) ) ) ) )
9513, 94syl5bb 250 . 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
) ) ) ) )
96 fsumvma2.3 . 2  |-  ( (
ph  /\  x  e.  ( 1 ... ( |_ `  A ) ) )  ->  B  e.  CC )
97 fsumvma2.4 . 2  |-  ( (
ph  /\  ( x  e.  ( 1 ... ( |_ `  A ) )  /\  (Λ `  x
)  =  0 ) )  ->  B  = 
0 )
981, 2, 5, 8, 95, 96, 97fsumvma 20998 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320    C_ wss 3321   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Fincfn 7110   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    x. cmul 8996    < clt 9121    <_ cle 9122    / cdiv 9678   NNcn 10001   2c2 10050   NN0cn0 10222   ZZcz 10283   ZZ>=cuz 10489   RR+crp 10613   [,]cicc 10920   ...cfz 11044   |_cfl 11202   ^cexp 11383   sum_csu 12480   Primecprime 13080   logclog 20453  Λcvma 20875
This theorem is referenced by:  chpval2  21003  rplogsumlem2  21180  rpvmasumlem  21182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ioc 10922  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-fl 11203  df-mod 11252  df-seq 11325  df-exp 11384  df-fac 11568  df-bc 11595  df-hash 11620  df-shft 11883  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-limsup 12266  df-clim 12283  df-rlim 12284  df-sum 12481  df-ef 12671  df-sin 12673  df-cos 12674  df-pi 12676  df-dvds 12854  df-gcd 13008  df-prm 13081  df-pc 13212  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-fbas 16700  df-fg 16701  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-ntr 17085  df-cls 17086  df-nei 17163  df-lp 17201  df-perf 17202  df-cn 17292  df-cnp 17293  df-haus 17380  df-tx 17595  df-hmeo 17788  df-fil 17879  df-fm 17971  df-flim 17972  df-flf 17973  df-xms 18351  df-ms 18352  df-tms 18353  df-cncf 18909  df-limc 19754  df-dv 19755  df-log 20455  df-vma 20881
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