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Theorem fsumvma 23211
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 5869 . . . . 5  |-  ( ^ `  z )  e.  _V
21a1i 11 . . . 4  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  e.  _V )
3 fveq2 5859 . . . . . . . 8  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( ^ `  <. p ,  k >. )
)
4 df-ov 6280 . . . . . . . 8  |-  ( p ^ k )  =  ( ^ `  <. p ,  k >. )
53, 4syl6eqr 2521 . . . . . . 7  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( p ^ k ) )
65eqeq2d 2476 . . . . . 6  |-  ( z  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  z
)  <->  x  =  (
p ^ k ) ) )
76biimpa 484 . . . . 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 3457 . . 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 465 . . . 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 620 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) )
1716simprd 463 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p ^ k )  e.  A )
1817ex 434 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
( p ^ k
)  e.  A ) )
1916simpld 459 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p  e.  Prime  /\  k  e.  NN ) )
2019simpld 459 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  p  e.  Prime )
2120adantrr 716 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  p  e.  Prime )
2219simprd 463 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  k  e.  NN )
2322adantrr 716 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  k  e.  NN )
2422ex 434 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
k  e.  NN ) )
2524ssrdv 3505 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  P )  ->  K  C_  NN )
2625sselda 3499 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  z  e.  K )  ->  z  e.  NN )
2726adantrl 715 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  z  e.  NN )
28 eqid 2462 . . . . . . . 8  |-  p  =  p
29 prmexpb 14108 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  ( p  =  p  /\  k  =  z ) ) )
3029baibd 902 . . . . . . . 8  |-  ( ( ( ( p  e. 
Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  /\  p  =  p )  ->  ( ( p ^
k )  =  ( p ^ z )  <-> 
k  =  z ) )
3128, 30mpan2 671 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3221, 21, 23, 27, 31syl22anc 1224 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3332ex 434 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
( k  e.  K  /\  z  e.  K
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) ) )
3418, 33dom2lem 7547 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  |->  ( p ^ k ) ) : K -1-1-> A
)
35 f1fi 7798 . . . 4  |-  ( ( A  e.  Fin  /\  ( k  e.  K  |->  ( p ^ k
) ) : K -1-1-> A )  ->  K  e.  Fin )
3613, 34, 35syl2anc 661 . . 3  |-  ( (
ph  /\  p  e.  P )  ->  K  e.  Fin )
3714simplbda 624 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p ^ k
)  e.  A )
38 fsumvma.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3938ralrimiva 2873 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
4039adantr 465 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  A. x  e.  A  B  e.  CC )
418eleq1d 2531 . . . . 5  |-  ( x  =  ( p ^
k )  ->  ( B  e.  CC  <->  C  e.  CC ) )
4241rspcv 3205 . . . 4  |-  ( ( p ^ k )  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  C  e.  CC ) )
4337, 40, 42sylc 60 . . 3  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  C  e.  CC )
4410, 11, 36, 43fsum2d 13537 . 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 2624 . . . 4  |-  F/_ y B
46 nfcsb1v 3446 . . . 4  |-  F/_ x [_ y  /  x ]_ B
47 csbeq1a 3439 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4845, 46, 47cbvsumi 13470 . . 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 3433 . . . 4  |-  ( y  =  ( ^ `  z )  ->  [_ y  /  x ]_ B  = 
[_ ( ^ `  z )  /  x ]_ B )
50 snfi 7588 . . . . . . 7  |-  { p }  e.  Fin
51 xpfi 7782 . . . . . . 7  |-  ( ( { p }  e.  Fin  /\  K  e.  Fin )  ->  ( { p }  X.  K )  e. 
Fin )
5250, 36, 51sylancr 663 . . . . . 6  |-  ( (
ph  /\  p  e.  P )  ->  ( { p }  X.  K )  e.  Fin )
5352ralrimiva 2873 . . . . 5  |-  ( ph  ->  A. p  e.  P  ( { p }  X.  K )  e.  Fin )
54 iunfi 7799 . . . . 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 661 . . . 4  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
56 fvex 5869 . . . . . . 7  |-  ( ^ `  a )  e.  _V
5756a1ii 27 . . . . . 6  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  _V ) )
58 eliunxp 5133 . . . . . . . . 9  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  <->  E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) ) )
5914simprbda 623 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
60 opelxp 5023 . . . . . . . . . . . . . 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 2534 . . . . . . . . . . . . 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 440 . . . . . . . . . . 11  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6564expimpd 603 . . . . . . . . . 10  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6665exlimdvv 1696 . . . . . . . . 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 3505 . . . . . . . . 9  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  C_  ( Prime  X.  NN ) )
6968sseld 3498 . . . . . . . 8  |-  ( ph  ->  ( b  e.  U_ p  e.  P  ( { p }  X.  K )  ->  b  e.  ( Prime  X.  NN ) ) )
7067, 69anim12d 563 . . . . . . 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 6813 . . . . . . . . . . 11  |-  ( a  e.  ( Prime  X.  NN )  ->  a  =  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7271fveq2d 5863 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. ) )
73 df-ov 6280 . . . . . . . . . 10  |-  ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7472, 73syl6eqr 2521 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ( 1st `  a
) ^ ( 2nd `  a ) ) )
75 1st2nd2 6813 . . . . . . . . . . 11  |-  ( b  e.  ( Prime  X.  NN )  ->  b  =  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7675fveq2d 5863 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. ) )
77 df-ov 6280 . . . . . . . . . 10  |-  ( ( 1st `  b ) ^ ( 2nd `  b
) )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7876, 77syl6eqr 2521 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) )
7974, 78eqeqan12d 2485 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  ( ( 1st `  a ) ^
( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) ) )
80 xp1st 6806 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 1st `  a
)  e.  Prime )
81 xp2nd 6807 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 2nd `  a
)  e.  NN )
8280, 81jca 532 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ( 1st `  a )  e.  Prime  /\  ( 2nd `  a
)  e.  NN ) )
83 xp1st 6806 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 1st `  b
)  e.  Prime )
84 xp2nd 6807 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 2nd `  b
)  e.  NN )
8583, 84jca 532 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ( 1st `  b )  e.  Prime  /\  ( 2nd `  b
)  e.  NN ) )
86 prmexpb 14108 . . . . . . . . . 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 823 . . . . . . . . 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 477 . . . . . . . 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 6815 . . . . . . . 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 33 . . . . . 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 7547 . . . . 5  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V )
93 f1f1orn 5820 . . . . 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 5859 . . . . . 6  |-  ( a  =  z  ->  ( ^ `  a )  =  ( ^ `  z ) )
96 eqid 2462 . . . . . 6  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  =  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )
9795, 96, 1fvmpt 5943 . . . . 5  |-  ( z  e.  U_ p  e.  P  ( { p }  X.  K )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
9897adantl 466 . . . 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 5859 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( ^ `  <. p ,  k >. )
)
10099, 4syl6eqr 2521 . . . . . . . . . . . . . . 15  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( p ^ k ) )
101100eleq1d 2531 . . . . . . . . . . . . . 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 440 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  ( ^ `  a
)  e.  A ) )
104103expimpd 603 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
105104exlimdvv 1696 . . . . . . . . . 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 429 . . . . . . . 8  |-  ( (
ph  /\  a  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ^ `  a
)  e.  A )
108107, 96fmptd 6038 . . . . . . 7  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A )
109 frn 5730 . . . . . . 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 3499 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  y  e.  A
)
11246nfel1 2640 . . . . . . 7  |-  F/ x [_ y  /  x ]_ B  e.  CC
11347eleq1d 2531 . . . . . . 7  |-  ( x  =  y  ->  ( B  e.  CC  <->  [_ y  /  x ]_ B  e.  CC ) )
114112, 113rspc 3203 . . . . . 6  |-  ( y  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  [_ y  /  x ]_ B  e.  CC )
)
11539, 114mpan9 469 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  CC )
116111, 115syldan 470 . . . 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 13496 . . 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 2515 . 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 3499 . . . 4  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  x  e.  A
)
120119, 38syldan 470 . . 3  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  e.  CC )
121 eldif 3481 . . . . 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 5246 . . . . . . . . . 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 2476 . . . . . . . . . . 11  |-  ( a  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  a
)  <->  x  =  (
p ^ k ) ) )
124123rexiunxp 5136 . . . . . . . . . 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 461 . . . . . . . . . . . . . . . 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 2551 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( p ^ k
)  e.  A )
12914rbaibd 903 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p ^ k )  e.  A )  ->  (
( p  e.  P  /\  k  e.  K
)  <->  ( p  e. 
Prime  /\  k  e.  NN ) ) )
130129adantlr 714 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  (
p ^ k )  e.  A )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
131128, 130syldan 470 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
132131pm5.32da 641 . . . . . . . . . . . . 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 450 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  P  /\  k  e.  K )
) )
134 ancom 450 . . . . . . . . . . . . 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 1687 . . . . . . . . . . 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 2980 . . . . . . . . . . 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 2980 . . . . . . . . . . 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 3499 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  NN )
142 isppw2 23112 . . . . . . . . . . 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 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 2718 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =  0  <->  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
147146pm5.32da 641 . . . . . 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 434 . . . . . 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 429 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) ) )  ->  B  = 
0 )
153110, 120, 152, 12fsumss 13498 . 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 2510 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 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   _Vcvv 3108   [_csb 3430    \ cdif 3468    C_ wss 3471   {csn 4022   <.cop 4028   U_ciun 4320    |-> cmpt 4500    X. cxp 4992   ran crn 4995   -->wf 5577   -1-1->wf1 5578   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   1stc1st 6774   2ndc2nd 6775   Fincfn 7508   CCcc 9481   0cc0 9483   NNcn 10527   ^cexp 12124   sum_csu 13459   Primecprime 14067  Λcvma 23088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-fi 7862  df-sup 7892  df-oi 7926  df-card 8311  df-cda 8539  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-q 11174  df-rp 11212  df-xneg 11309  df-xadd 11310  df-xmul 11311  df-ioo 11524  df-ioc 11525  df-ico 11526  df-icc 11527  df-fz 11664  df-fzo 11784  df-fl 11888  df-mod 11955  df-seq 12066  df-exp 12125  df-fac 12311  df-bc 12338  df-hash 12363  df-shft 12852  df-cj 12884  df-re 12885  df-im 12886  df-sqr 13020  df-abs 13021  df-limsup 13245  df-clim 13262  df-rlim 13263  df-sum 13460  df-ef 13656  df-sin 13658  df-cos 13659  df-pi 13661  df-dvds 13839  df-gcd 13995  df-prm 14068  df-pc 14211  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-starv 14561  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-unif 14569  df-hom 14570  df-cco 14571  df-rest 14669  df-topn 14670  df-0g 14688  df-gsum 14689  df-topgen 14690  df-pt 14691  df-prds 14694  df-xrs 14748  df-qtop 14753  df-imas 14754  df-xps 14756  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-submnd 15773  df-mulg 15856  df-cntz 16145  df-cmn 16591  df-psmet 18177  df-xmet 18178  df-met 18179  df-bl 18180  df-mopn 18181  df-fbas 18182  df-fg 18183  df-cnfld 18187  df-top 19161  df-bases 19163  df-topon 19164  df-topsp 19165  df-cld 19281  df-ntr 19282  df-cls 19283  df-nei 19360  df-lp 19398  df-perf 19399  df-cn 19489  df-cnp 19490  df-haus 19577  df-tx 19793  df-hmeo 19986  df-fil 20077  df-fm 20169  df-flim 20170  df-flf 20171  df-xms 20553  df-ms 20554  df-tms 20555  df-cncf 21112  df-limc 22000  df-dv 22001  df-log 22667  df-vma 23094
This theorem is referenced by:  fsumvma2  23212  vmasum  23214
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