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

Theorem fsumvma 23869
Description: Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
Hypotheses
Ref Expression
fsumvma.1  |-  ( x  =  ( p ^
k )  ->  B  =  C )
fsumvma.2  |-  ( ph  ->  A  e.  Fin )
fsumvma.3  |-  ( ph  ->  A  C_  NN )
fsumvma.4  |-  ( ph  ->  P  e.  Fin )
fsumvma.5  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  (
p ^ k )  e.  A ) ) )
fsumvma.6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
fsumvma.7  |-  ( (
ph  /\  ( x  e.  A  /\  (Λ `  x )  =  0 ) )  ->  B  =  0 )
Assertion
Ref Expression
fsumvma  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ p  e.  P  sum_ k  e.  K  C )
Distinct variable groups:    k, p, x, A    x, C    k, K, x    ph, k, p, x    B, k, p    P, k, p, x
Allowed substitution hints:    B( x)    C( k, p)    K( p)

Proof of Theorem fsumvma
Dummy variables  a 
z  b  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5859 . . . . 5  |-  ( ^ `  z )  e.  _V
21a1i 11 . . . 4  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  e.  _V )
3 fveq2 5849 . . . . . . . 8  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( ^ `  <. p ,  k >. )
)
4 df-ov 6281 . . . . . . . 8  |-  ( p ^ k )  =  ( ^ `  <. p ,  k >. )
53, 4syl6eqr 2461 . . . . . . 7  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( p ^ k ) )
65eqeq2d 2416 . . . . . 6  |-  ( z  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  z
)  <->  x  =  (
p ^ k ) ) )
76biimpa 482 . . . . 5  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  x  =  ( p ^ k ) )
8 fsumvma.1 . . . . 5  |-  ( x  =  ( p ^
k )  ->  B  =  C )
97, 8syl 17 . . . 4  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  B  =  C )
102, 9csbied 3400 . . 3  |-  ( z  =  <. p ,  k
>.  ->  [_ ( ^ `  z )  /  x ]_ B  =  C
)
11 fsumvma.4 . . 3  |-  ( ph  ->  P  e.  Fin )
12 fsumvma.2 . . . . 5  |-  ( ph  ->  A  e.  Fin )
1312adantr 463 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  A  e.  Fin )
14 fsumvma.5 . . . . . . . . 9  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  (
p ^ k )  e.  A ) ) )
1514biimpd 207 . . . . . . . 8  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) ) )
1615impl 618 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) )
1716simprd 461 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p ^ k )  e.  A )
1817ex 432 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
( p ^ k
)  e.  A ) )
1916simpld 457 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p  e.  Prime  /\  k  e.  NN ) )
2019simpld 457 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  p  e.  Prime )
2120adantrr 715 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  p  e.  Prime )
2219simprd 461 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  k  e.  NN )
2322adantrr 715 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  k  e.  NN )
2422ex 432 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
k  e.  NN ) )
2524ssrdv 3448 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  P )  ->  K  C_  NN )
2625sselda 3442 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  z  e.  K )  ->  z  e.  NN )
2726adantrl 714 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  z  e.  NN )
28 eqid 2402 . . . . . . . 8  |-  p  =  p
29 prmexpb 14467 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  ( p  =  p  /\  k  =  z ) ) )
3029baibd 910 . . . . . . . 8  |-  ( ( ( ( p  e. 
Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  /\  p  =  p )  ->  ( ( p ^
k )  =  ( p ^ z )  <-> 
k  =  z ) )
3128, 30mpan2 669 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3221, 21, 23, 27, 31syl22anc 1231 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3332ex 432 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
( k  e.  K  /\  z  e.  K
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) ) )
3418, 33dom2lem 7593 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  |->  ( p ^ k ) ) : K -1-1-> A
)
35 f1fi 7841 . . . 4  |-  ( ( A  e.  Fin  /\  ( k  e.  K  |->  ( p ^ k
) ) : K -1-1-> A )  ->  K  e.  Fin )
3613, 34, 35syl2anc 659 . . 3  |-  ( (
ph  /\  p  e.  P )  ->  K  e.  Fin )
3714simplbda 622 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p ^ k
)  e.  A )
38 fsumvma.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3938ralrimiva 2818 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
4039adantr 463 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  A. x  e.  A  B  e.  CC )
418eleq1d 2471 . . . . 5  |-  ( x  =  ( p ^
k )  ->  ( B  e.  CC  <->  C  e.  CC ) )
4241rspcv 3156 . . . 4  |-  ( ( p ^ k )  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  C  e.  CC ) )
4337, 40, 42sylc 59 . . 3  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  C  e.  CC )
4410, 11, 36, 43fsum2d 13737 . 2  |-  ( ph  -> 
sum_ p  e.  P  sum_ k  e.  K  C  =  sum_ z  e.  U_  p  e.  P  ( { p }  X.  K ) [_ ( ^ `  z )  /  x ]_ B )
45 nfcv 2564 . . . 4  |-  F/_ y B
46 nfcsb1v 3389 . . . 4  |-  F/_ x [_ y  /  x ]_ B
47 csbeq1a 3382 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4845, 46, 47cbvsumi 13668 . . 3  |-  sum_ x  e.  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) B  =  sum_ y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) [_ y  /  x ]_ B
49 csbeq1 3376 . . . 4  |-  ( y  =  ( ^ `  z )  ->  [_ y  /  x ]_ B  = 
[_ ( ^ `  z )  /  x ]_ B )
50 snfi 7634 . . . . . . 7  |-  { p }  e.  Fin
51 xpfi 7825 . . . . . . 7  |-  ( ( { p }  e.  Fin  /\  K  e.  Fin )  ->  ( { p }  X.  K )  e. 
Fin )
5250, 36, 51sylancr 661 . . . . . 6  |-  ( (
ph  /\  p  e.  P )  ->  ( { p }  X.  K )  e.  Fin )
5352ralrimiva 2818 . . . . 5  |-  ( ph  ->  A. p  e.  P  ( { p }  X.  K )  e.  Fin )
54 iunfi 7842 . . . . 5  |-  ( ( P  e.  Fin  /\  A. p  e.  P  ( { p }  X.  K )  e.  Fin )  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
5511, 53, 54syl2anc 659 . . . 4  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
56 fvex 5859 . . . . . . 7  |-  ( ^ `  a )  e.  _V
5756a1ii 12 . . . . . 6  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  _V ) )
58 eliunxp 4961 . . . . . . . . 9  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  <->  E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) ) )
5914simprbda 621 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
60 opelxp 4853 . . . . . . . . . . . . . 14  |-  ( <.
p ,  k >.  e.  ( Prime  X.  NN ) 
<->  ( p  e.  Prime  /\  k  e.  NN ) )
6159, 60sylibr 212 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  <. p ,  k >.  e.  ( Prime  X.  NN ) )
62 eleq1 2474 . . . . . . . . . . . . 13  |-  ( a  =  <. p ,  k
>.  ->  ( a  e.  ( Prime  X.  NN ) 
<-> 
<. p ,  k >.  e.  ( Prime  X.  NN ) ) )
6361, 62syl5ibrcom 222 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  a  e.  ( Prime  X.  NN ) ) )
6463impancom 438 . . . . . . . . . . 11  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6564expimpd 601 . . . . . . . . . 10  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6665exlimdvv 1746 . . . . . . . . 9  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6758, 66syl5bi 217 . . . . . . . 8  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6867ssrdv 3448 . . . . . . . . 9  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  C_  ( Prime  X.  NN ) )
6968sseld 3441 . . . . . . . 8  |-  ( ph  ->  ( b  e.  U_ p  e.  P  ( { p }  X.  K )  ->  b  e.  ( Prime  X.  NN ) ) )
7067, 69anim12d 561 . . . . . . 7  |-  ( ph  ->  ( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  /\  b  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) ) ) )
71 1st2nd2 6821 . . . . . . . . . . 11  |-  ( a  e.  ( Prime  X.  NN )  ->  a  =  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7271fveq2d 5853 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. ) )
73 df-ov 6281 . . . . . . . . . 10  |-  ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7472, 73syl6eqr 2461 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ( 1st `  a
) ^ ( 2nd `  a ) ) )
75 1st2nd2 6821 . . . . . . . . . . 11  |-  ( b  e.  ( Prime  X.  NN )  ->  b  =  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7675fveq2d 5853 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. ) )
77 df-ov 6281 . . . . . . . . . 10  |-  ( ( 1st `  b ) ^ ( 2nd `  b
) )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7876, 77syl6eqr 2461 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) )
7974, 78eqeqan12d 2425 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  ( ( 1st `  a ) ^
( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) ) )
80 xp1st 6814 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 1st `  a
)  e.  Prime )
81 xp2nd 6815 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 2nd `  a
)  e.  NN )
8280, 81jca 530 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ( 1st `  a )  e.  Prime  /\  ( 2nd `  a
)  e.  NN ) )
83 xp1st 6814 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 1st `  b
)  e.  Prime )
84 xp2nd 6815 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 2nd `  b
)  e.  NN )
8583, 84jca 530 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ( 1st `  b )  e.  Prime  /\  ( 2nd `  b
)  e.  NN ) )
86 prmexpb 14467 . . . . . . . . . 10  |-  ( ( ( ( 1st `  a
)  e.  Prime  /\  ( 1st `  b )  e. 
Prime )  /\  (
( 2nd `  a
)  e.  NN  /\  ( 2nd `  b )  e.  NN ) )  ->  ( ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) )  <->  ( ( 1st `  a )  =  ( 1st `  b
)  /\  ( 2nd `  a )  =  ( 2nd `  b ) ) ) )
8786an4s 827 . . . . . . . . 9  |-  ( ( ( ( 1st `  a
)  e.  Prime  /\  ( 2nd `  a )  e.  NN )  /\  (
( 1st `  b
)  e.  Prime  /\  ( 2nd `  b )  e.  NN ) )  -> 
( ( ( 1st `  a ) ^ ( 2nd `  a ) )  =  ( ( 1st `  b ) ^ ( 2nd `  b ) )  <-> 
( ( 1st `  a
)  =  ( 1st `  b )  /\  ( 2nd `  a )  =  ( 2nd `  b
) ) ) )
8882, 85, 87syl2an 475 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ( 1st `  a
) ^ ( 2nd `  a ) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) )  <->  ( ( 1st `  a )  =  ( 1st `  b
)  /\  ( 2nd `  a )  =  ( 2nd `  b ) ) ) )
89 xpopth 6823 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ( 1st `  a
)  =  ( 1st `  b )  /\  ( 2nd `  a )  =  ( 2nd `  b
) )  <->  a  =  b ) )
9079, 88, 893bitrd 279 . . . . . . 7  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  a  =  b ) )
9170, 90syl6 31 . . . . . 6  |-  ( ph  ->  ( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  /\  b  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ( ^ `  a )  =  ( ^ `  b )  <-> 
a  =  b ) ) )
9257, 91dom2lem 7593 . . . . 5  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V )
93 f1f1orn 5810 . . . . 5  |-  ( ( a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V  ->  ( a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-onto-> ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )
9492, 93syl 17 . . . 4  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-onto-> ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )
95 fveq2 5849 . . . . . 6  |-  ( a  =  z  ->  ( ^ `  a )  =  ( ^ `  z ) )
96 eqid 2402 . . . . . 6  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  =  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )
9795, 96, 1fvmpt 5932 . . . . 5  |-  ( z  e.  U_ p  e.  P  ( { p }  X.  K )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
9897adantl 464 . . . 4  |-  ( (
ph  /\  z  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
99 fveq2 5849 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( ^ `  <. p ,  k >. )
)
10099, 4syl6eqr 2461 . . . . . . . . . . . . . . 15  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( p ^ k ) )
101100eleq1d 2471 . . . . . . . . . . . . . 14  |-  ( a  =  <. p ,  k
>.  ->  ( ( ^ `  a )  e.  A  <->  ( p ^ k )  e.  A ) )
10237, 101syl5ibrcom 222 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  ( ^ `  a )  e.  A ) )
103102impancom 438 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  ( ^ `  a
)  e.  A ) )
104103expimpd 601 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
105104exlimdvv 1746 . . . . . . . . . 10  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
10658, 105syl5bi 217 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  A ) )
107106imp 427 . . . . . . . 8  |-  ( (
ph  /\  a  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ^ `  a
)  e.  A )
108107, 96fmptd 6033 . . . . . . 7  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A )
109 frn 5720 . . . . . . 7  |-  ( ( a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A  ->  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  C_  A )
110108, 109syl 17 . . . . . 6  |-  ( ph  ->  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  C_  A )
111110sselda 3442 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  y  e.  A
)
11246nfel1 2580 . . . . . . 7  |-  F/ x [_ y  /  x ]_ B  e.  CC
11347eleq1d 2471 . . . . . . 7  |-  ( x  =  y  ->  ( B  e.  CC  <->  [_ y  /  x ]_ B  e.  CC ) )
114112, 113rspc 3154 . . . . . 6  |-  ( y  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  [_ y  /  x ]_ B  e.  CC )
)
11539, 114mpan9 467 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  CC )
116111, 115syldan 468 . . . 4  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  [_ y  /  x ]_ B  e.  CC )
11749, 55, 94, 98, 116fsumf1o 13694 . . 3  |-  ( ph  -> 
sum_ y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) [_ y  /  x ]_ B  =  sum_ z  e.  U_  p  e.  P  ( { p }  X.  K ) [_ ( ^ `  z )  /  x ]_ B )
11848, 117syl5eq 2455 . 2  |-  ( ph  -> 
sum_ x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) B  =  sum_ z  e.  U_  p  e.  P  ( { p }  X.  K ) [_ ( ^ `  z )  /  x ]_ B )
119110sselda 3442 . . . 4  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  x  e.  A
)
120119, 38syldan 468 . . 3  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  e.  CC )
121 eldif 3424 . . . . 5  |-  ( x  e.  ( A  \  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  <-> 
( x  e.  A  /\  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
12296, 56elrnmpti 5074 . . . . . . . . . 10  |-  ( x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  <->  E. a  e.  U_  p  e.  P  ( { p }  X.  K ) x  =  ( ^ `  a
) )
123100eqeq2d 2416 . . . . . . . . . . 11  |-  ( a  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  a
)  <->  x  =  (
p ^ k ) ) )
124123rexiunxp 4964 . . . . . . . . . 10  |-  ( E. a  e.  U_  p  e.  P  ( {
p }  X.  K
) x  =  ( ^ `  a )  <->  E. p  e.  P  E. k  e.  K  x  =  ( p ^ k ) )
125122, 124bitri 249 . . . . . . . . 9  |-  ( x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  <->  E. p  e.  P  E. k  e.  K  x  =  ( p ^ k ) )
126 simpr 459 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  =  ( p ^ k ) )
127 simplr 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  e.  A )
128126, 127eqeltrrd 2491 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( p ^ k
)  e.  A )
12914rbaibd 911 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p ^ k )  e.  A )  ->  (
( p  e.  P  /\  k  e.  K
)  <->  ( p  e. 
Prime  /\  k  e.  NN ) ) )
130129adantlr 713 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  (
p ^ k )  e.  A )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
131128, 130syldan 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
132131pm5.32da 639 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  =  ( p ^ k )  /\  ( p  e.  P  /\  k  e.  K ) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  Prime  /\  k  e.  NN ) ) ) )
133 ancom 448 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  P  /\  k  e.  K )
) )
134 ancom 448 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^ k ) )  <->  ( x  =  ( p ^ k
)  /\  ( p  e.  Prime  /\  k  e.  NN ) ) )
135132, 133, 1343bitr4g 288 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( p  e.  P  /\  k  e.  K )  /\  x  =  ( p ^
k ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^
k ) ) ) )
1361352exbidv 1737 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p E. k ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  E. p E. k ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^
k ) ) ) )
137 r2ex 2930 . . . . . . . . . . 11  |-  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^
k )  <->  E. p E. k ( ( p  e.  P  /\  k  e.  K )  /\  x  =  ( p ^
k ) ) )
138 r2ex 2930 . . . . . . . . . . 11  |-  ( E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^ k )  <->  E. p E. k ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^ k ) ) )
139136, 137, 1383bitr4g 288 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^ k )  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^ k ) ) )
140 fsumvma.3 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  NN )
141140sselda 3442 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  NN )
142 isppw2 23770 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
143141, 142syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
144139, 143bitr4d 256 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^ k )  <->  (Λ `  x
)  =/=  0 ) )
145125, 144syl5bb 257 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) )  <->  (Λ `  x
)  =/=  0 ) )
146145necon2bbid 2659 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =  0  <->  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
147146pm5.32da 639 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  (Λ `  x
)  =  0 )  <-> 
( x  e.  A  /\  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) ) )
148 fsumvma.7 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  A  /\  (Λ `  x )  =  0 ) )  ->  B  =  0 )
149148ex 432 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  (Λ `  x
)  =  0 )  ->  B  =  0 ) )
150147, 149sylbird 235 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
151121, 150syl5bi 217 . . . 4  |-  ( ph  ->  ( x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
152151imp 427 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) ) )  ->  B  = 
0 )
153110, 120, 152, 12fsumss 13696 . 2  |-  ( ph  -> 
sum_ x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) B  =  sum_ x  e.  A  B )
15444, 118, 1533eqtr2rd 2450 1  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ p  e.  P  sum_ k  e.  K  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   A.wral 2754   E.wrex 2755   _Vcvv 3059   [_csb 3373    \ cdif 3411    C_ wss 3414   {csn 3972   <.cop 3978   U_ciun 4271    |-> cmpt 4453    X. cxp 4821   ran crn 4824   -->wf 5565   -1-1->wf1 5566   -1-1-onto->wf1o 5568   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783   Fincfn 7554   CCcc 9520   0cc0 9522   NNcn 10576   ^cexp 12210   sum_csu 13657   Primecprime 14426  Λcvma 23746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ioc 11587  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-mod 12035  df-seq 12152  df-exp 12211  df-fac 12398  df-bc 12425  df-hash 12453  df-shft 13049  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-limsup 13443  df-clim 13460  df-rlim 13461  df-sum 13658  df-ef 14012  df-sin 14014  df-cos 14015  df-pi 14017  df-dvds 14196  df-gcd 14354  df-prm 14427  df-pc 14570  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563  df-log 23236  df-vma 23752
This theorem is referenced by:  fsumvma2  23870  vmasum  23872
  Copyright terms: Public domain W3C validator