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Theorem fsumvma 20998
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 5743 . . . . 5  |-  ( ^ `  z )  e.  _V
21a1i 11 . . . 4  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  e.  _V )
3 fveq2 5729 . . . . . . . 8  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( ^ `  <. p ,  k >. )
)
4 df-ov 6085 . . . . . . . 8  |-  ( p ^ k )  =  ( ^ `  <. p ,  k >. )
53, 4syl6eqr 2487 . . . . . . 7  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( p ^ k ) )
65eqeq2d 2448 . . . . . 6  |-  ( z  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  z
)  <->  x  =  (
p ^ k ) ) )
76biimpa 472 . . . . 5  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  x  =  ( p ^ k ) )
8 fsumvma.1 . . . . 5  |-  ( x  =  ( p ^
k )  ->  B  =  C )
97, 8syl 16 . . . 4  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  B  =  C )
102, 9csbied 3294 . . 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 453 . . . 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 200 . . . . . . . 8  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) ) )
1615impl 605 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) )
1716simprd 451 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p ^ k )  e.  A )
1817ex 425 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
( p ^ k
)  e.  A ) )
1916simpld 447 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p  e.  Prime  /\  k  e.  NN ) )
2019simpld 447 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  p  e.  Prime )
2120adantrr 699 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  p  e.  Prime )
2219simprd 451 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  k  e.  NN )
2322adantrr 699 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  k  e.  NN )
2422ex 425 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
k  e.  NN ) )
2524ssrdv 3355 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  P )  ->  K  C_  NN )
2625sselda 3349 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  z  e.  K )  ->  z  e.  NN )
2726adantrl 698 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  z  e.  NN )
28 eqid 2437 . . . . . . . 8  |-  p  =  p
29 prmexpb 13118 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  ( p  =  p  /\  k  =  z ) ) )
3029baibd 877 . . . . . . . 8  |-  ( ( ( ( p  e. 
Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  /\  p  =  p )  ->  ( ( p ^
k )  =  ( p ^ z )  <-> 
k  =  z ) )
3128, 30mpan2 654 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3221, 21, 23, 27, 31syl22anc 1186 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3332ex 425 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
( k  e.  K  /\  z  e.  K
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) ) )
3418, 33dom2lem 7148 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  |->  ( p ^ k ) ) : K -1-1-> A
)
35 f1fi 7394 . . . 4  |-  ( ( A  e.  Fin  /\  ( k  e.  K  |->  ( p ^ k
) ) : K -1-1-> A )  ->  K  e.  Fin )
3613, 34, 35syl2anc 644 . . 3  |-  ( (
ph  /\  p  e.  P )  ->  K  e.  Fin )
3714simplbda 609 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p ^ k
)  e.  A )
38 fsumvma.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3938ralrimiva 2790 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
4039adantr 453 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  A. x  e.  A  B  e.  CC )
418eleq1d 2503 . . . . 5  |-  ( x  =  ( p ^
k )  ->  ( B  e.  CC  <->  C  e.  CC ) )
4241rspcv 3049 . . . 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 12556 . 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 2573 . . . 4  |-  F/_ y B
46 nfcsb1v 3284 . . . 4  |-  F/_ x [_ y  /  x ]_ B
47 csbeq1a 3260 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4845, 46, 47cbvsumi 12492 . . 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 3255 . . . 4  |-  ( y  =  ( ^ `  z )  ->  [_ y  /  x ]_ B  = 
[_ ( ^ `  z )  /  x ]_ B )
50 snfi 7188 . . . . . . 7  |-  { p }  e.  Fin
51 xpfi 7379 . . . . . . 7  |-  ( ( { p }  e.  Fin  /\  K  e.  Fin )  ->  ( { p }  X.  K )  e. 
Fin )
5250, 36, 51sylancr 646 . . . . . 6  |-  ( (
ph  /\  p  e.  P )  ->  ( { p }  X.  K )  e.  Fin )
5352ralrimiva 2790 . . . . 5  |-  ( ph  ->  A. p  e.  P  ( { p }  X.  K )  e.  Fin )
54 iunfi 7395 . . . . 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 644 . . . 4  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
56 fvex 5743 . . . . . . 7  |-  ( ^ `  a )  e.  _V
5756a1ii 26 . . . . . 6  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  _V ) )
58 eliunxp 5013 . . . . . . . . 9  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  <->  E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) ) )
5914simprbda 608 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
60 opelxp 4909 . . . . . . . . . . . . . 14  |-  ( <.
p ,  k >.  e.  ( Prime  X.  NN ) 
<->  ( p  e.  Prime  /\  k  e.  NN ) )
6159, 60sylibr 205 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  <. p ,  k >.  e.  ( Prime  X.  NN ) )
62 eleq1 2497 . . . . . . . . . . . . 13  |-  ( a  =  <. p ,  k
>.  ->  ( a  e.  ( Prime  X.  NN ) 
<-> 
<. p ,  k >.  e.  ( Prime  X.  NN ) ) )
6361, 62syl5ibrcom 215 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  a  e.  ( Prime  X.  NN ) ) )
6463impancom 429 . . . . . . . . . . 11  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6564expimpd 588 . . . . . . . . . 10  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6665exlimdvv 1648 . . . . . . . . 9  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6758, 66syl5bi 210 . . . . . . . 8  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6867ssrdv 3355 . . . . . . . . 9  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  C_  ( Prime  X.  NN ) )
6968sseld 3348 . . . . . . . 8  |-  ( ph  ->  ( b  e.  U_ p  e.  P  ( { p }  X.  K )  ->  b  e.  ( Prime  X.  NN ) ) )
7067, 69anim12d 548 . . . . . . 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 6387 . . . . . . . . . . 11  |-  ( a  e.  ( Prime  X.  NN )  ->  a  =  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7271fveq2d 5733 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. ) )
73 df-ov 6085 . . . . . . . . . 10  |-  ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7472, 73syl6eqr 2487 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ( 1st `  a
) ^ ( 2nd `  a ) ) )
75 1st2nd2 6387 . . . . . . . . . . 11  |-  ( b  e.  ( Prime  X.  NN )  ->  b  =  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7675fveq2d 5733 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. ) )
77 df-ov 6085 . . . . . . . . . 10  |-  ( ( 1st `  b ) ^ ( 2nd `  b
) )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7876, 77syl6eqr 2487 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) )
7974, 78eqeqan12d 2452 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  ( ( 1st `  a ) ^
( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) ) )
80 xp1st 6377 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 1st `  a
)  e.  Prime )
81 xp2nd 6378 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 2nd `  a
)  e.  NN )
8280, 81jca 520 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ( 1st `  a )  e.  Prime  /\  ( 2nd `  a
)  e.  NN ) )
83 xp1st 6377 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 1st `  b
)  e.  Prime )
84 xp2nd 6378 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 2nd `  b
)  e.  NN )
8583, 84jca 520 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ( 1st `  b )  e.  Prime  /\  ( 2nd `  b
)  e.  NN ) )
86 prmexpb 13118 . . . . . . . . . 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 801 . . . . . . . . 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 465 . . . . . . . 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 6389 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ( 1st `  a
)  =  ( 1st `  b )  /\  ( 2nd `  a )  =  ( 2nd `  b
) )  <->  a  =  b ) )
9079, 88, 893bitrd 272 . . . . . . 7  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  a  =  b ) )
9170, 90syl6 32 . . . . . 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 7148 . . . . 5  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V )
93 f1f1orn 5686 . . . . 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 16 . . . 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 5729 . . . . . 6  |-  ( a  =  z  ->  ( ^ `  a )  =  ( ^ `  z ) )
96 eqid 2437 . . . . . 6  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  =  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )
9795, 96, 1fvmpt 5807 . . . . 5  |-  ( z  e.  U_ p  e.  P  ( { p }  X.  K )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
9897adantl 454 . . . 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 5729 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( ^ `  <. p ,  k >. )
)
10099, 4syl6eqr 2487 . . . . . . . . . . . . . . 15  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( p ^ k ) )
101100eleq1d 2503 . . . . . . . . . . . . . 14  |-  ( a  =  <. p ,  k
>.  ->  ( ( ^ `  a )  e.  A  <->  ( p ^ k )  e.  A ) )
10237, 101syl5ibrcom 215 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  ( ^ `  a )  e.  A ) )
103102impancom 429 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  ( ^ `  a
)  e.  A ) )
104103expimpd 588 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
105104exlimdvv 1648 . . . . . . . . . 10  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
10658, 105syl5bi 210 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  A ) )
107106imp 420 . . . . . . . 8  |-  ( (
ph  /\  a  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ^ `  a
)  e.  A )
108107, 96fmptd 5894 . . . . . . 7  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A )
109 frn 5598 . . . . . . 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 16 . . . . . 6  |-  ( ph  ->  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  C_  A )
111110sselda 3349 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  y  e.  A
)
11246nfel1 2583 . . . . . . 7  |-  F/ x [_ y  /  x ]_ B  e.  CC
11347eleq1d 2503 . . . . . . 7  |-  ( x  =  y  ->  ( B  e.  CC  <->  [_ y  /  x ]_ B  e.  CC ) )
114112, 113rspc 3047 . . . . . 6  |-  ( y  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  [_ y  /  x ]_ B  e.  CC )
)
11539, 114mpan9 457 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  CC )
116111, 115syldan 458 . . . 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 12518 . . 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 2481 . 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 3349 . . . 4  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  x  e.  A
)
120119, 38syldan 458 . . 3  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  e.  CC )
121 eldif 3331 . . . . 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 5122 . . . . . . . . . 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 2448 . . . . . . . . . . 11  |-  ( a  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  a
)  <->  x  =  (
p ^ k ) ) )
124123rexiunxp 5016 . . . . . . . . . 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 242 . . . . . . . . 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 449 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  =  ( p ^ k ) )
127 simplr 733 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  e.  A )
128126, 127eqeltrrd 2512 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( p ^ k
)  e.  A )
12914rbaibd 878 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p ^ k )  e.  A )  ->  (
( p  e.  P  /\  k  e.  K
)  <->  ( p  e. 
Prime  /\  k  e.  NN ) ) )
130129adantlr 697 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  (
p ^ k )  e.  A )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
131128, 130syldan 458 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
132131pm5.32da 624 . . . . . . . . . . . . 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 439 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  P  /\  k  e.  K )
) )
134 ancom 439 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^ k ) )  <->  ( x  =  ( p ^ k
)  /\  ( p  e.  Prime  /\  k  e.  NN ) ) )
135132, 133, 1343bitr4g 281 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( p  e.  P  /\  k  e.  K )  /\  x  =  ( p ^
k ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^
k ) ) ) )
1361352exbidv 1639 . . . . . . . . . . 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 2744 . . . . . . . . . . 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 2744 . . . . . . . . . . 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 281 . . . . . . . . . 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 3349 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  NN )
142 isppw2 20899 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
143141, 142syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
144139, 143bitr4d 249 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^ k )  <->  (Λ `  x
)  =/=  0 ) )
145125, 144syl5bb 250 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) )  <->  (Λ `  x
)  =/=  0 ) )
146145necon2bbid 2663 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =  0  <->  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
147146pm5.32da 624 . . . . . 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 425 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  (Λ `  x
)  =  0 )  ->  B  =  0 ) )
150147, 149sylbird 228 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
151121, 150syl5bi 210 . . . 4  |-  ( ph  ->  ( x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
152151imp 420 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) ) )  ->  B  = 
0 )
153110, 120, 152, 12fsumss 12520 . 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 2476 1  |-  ( ph  -> 
sum_ x  e.  A  B  =  sum_ p  e.  P  sum_ k  e.  K  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707   _Vcvv 2957   [_csb 3252    \ cdif 3318    C_ wss 3321   {csn 3815   <.cop 3818   U_ciun 4094    e. cmpt 4267    X. cxp 4877   ran crn 4880   -->wf 5451   -1-1->wf1 5452   -1-1-onto->wf1o 5454   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349   Fincfn 7110   CCcc 8989   0cc0 8991   NNcn 10001   ^cexp 11383   sum_csu 12480   Primecprime 13080  Λcvma 20875
This theorem is referenced by:  fsumvma2  20999  vmasum  21001
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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