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Theorem fsumvma 24078
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 5830 . . . . 5  |-  ( ^ `  z )  e.  _V
21a1i 11 . . . 4  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  e.  _V )
3 fveq2 5820 . . . . . . . 8  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( ^ `  <. p ,  k >. )
)
4 df-ov 6247 . . . . . . . 8  |-  ( p ^ k )  =  ( ^ `  <. p ,  k >. )
53, 4syl6eqr 2475 . . . . . . 7  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( p ^ k ) )
65eqeq2d 2433 . . . . . 6  |-  ( z  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  z
)  <->  x  =  (
p ^ k ) ) )
76biimpa 486 . . . . 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 3360 . . 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 466 . . . 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 210 . . . . . . . 8  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) ) )
1615impl 624 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) )
1716simprd 464 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p ^ k )  e.  A )
1817ex 435 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
( p ^ k
)  e.  A ) )
1916simpld 460 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p  e.  Prime  /\  k  e.  NN ) )
2019simpld 460 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  p  e.  Prime )
2120adantrr 721 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  p  e.  Prime )
2219simprd 464 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  k  e.  NN )
2322adantrr 721 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  k  e.  NN )
2422ex 435 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
k  e.  NN ) )
2524ssrdv 3408 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  P )  ->  K  C_  NN )
2625sselda 3402 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  z  e.  K )  ->  z  e.  NN )
2726adantrl 720 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  z  e.  NN )
28 eqid 2423 . . . . . . . 8  |-  p  =  p
29 prmexpb 14608 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  ( p  =  p  /\  k  =  z ) ) )
3029baibd 917 . . . . . . . 8  |-  ( ( ( ( p  e. 
Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  /\  p  =  p )  ->  ( ( p ^
k )  =  ( p ^ z )  <-> 
k  =  z ) )
3128, 30mpan2 675 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3221, 21, 23, 27, 31syl22anc 1265 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3332ex 435 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
( k  e.  K  /\  z  e.  K
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) ) )
3418, 33dom2lem 7558 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  |->  ( p ^ k ) ) : K -1-1-> A
)
35 f1fi 7809 . . . 4  |-  ( ( A  e.  Fin  /\  ( k  e.  K  |->  ( p ^ k
) ) : K -1-1-> A )  ->  K  e.  Fin )
3613, 34, 35syl2anc 665 . . 3  |-  ( (
ph  /\  p  e.  P )  ->  K  e.  Fin )
3714simplbda 628 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p ^ k
)  e.  A )
38 fsumvma.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3938ralrimiva 2774 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
4039adantr 466 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  A. x  e.  A  B  e.  CC )
418eleq1d 2485 . . . . 5  |-  ( x  =  ( p ^
k )  ->  ( B  e.  CC  <->  C  e.  CC ) )
4241rspcv 3116 . . . 4  |-  ( ( p ^ k )  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  C  e.  CC ) )
4337, 40, 42sylc 62 . . 3  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  C  e.  CC )
4410, 11, 36, 43fsum2d 13770 . 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 3349 . . . 4  |-  F/_ x [_ y  /  x ]_ B
47 csbeq1a 3342 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4845, 46, 47cbvsumi 13701 . . 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 3336 . . . 4  |-  ( y  =  ( ^ `  z )  ->  [_ y  /  x ]_ B  = 
[_ ( ^ `  z )  /  x ]_ B )
50 snfi 7599 . . . . . . 7  |-  { p }  e.  Fin
51 xpfi 7790 . . . . . . 7  |-  ( ( { p }  e.  Fin  /\  K  e.  Fin )  ->  ( { p }  X.  K )  e. 
Fin )
5250, 36, 51sylancr 667 . . . . . 6  |-  ( (
ph  /\  p  e.  P )  ->  ( { p }  X.  K )  e.  Fin )
5352ralrimiva 2774 . . . . 5  |-  ( ph  ->  A. p  e.  P  ( { p }  X.  K )  e.  Fin )
54 iunfi 7810 . . . . 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 665 . . . 4  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
56 fvex 5830 . . . . . . 7  |-  ( ^ `  a )  e.  _V
57562a1i 12 . . . . . 6  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  _V ) )
58 eliunxp 4929 . . . . . . . . 9  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  <->  E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) ) )
5914simprbda 627 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
60 opelxp 4821 . . . . . . . . . . . . . 14  |-  ( <.
p ,  k >.  e.  ( Prime  X.  NN ) 
<->  ( p  e.  Prime  /\  k  e.  NN ) )
6159, 60sylibr 215 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  <. p ,  k >.  e.  ( Prime  X.  NN ) )
62 eleq1 2489 . . . . . . . . . . . . 13  |-  ( a  =  <. p ,  k
>.  ->  ( a  e.  ( Prime  X.  NN ) 
<-> 
<. p ,  k >.  e.  ( Prime  X.  NN ) ) )
6361, 62syl5ibrcom 225 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  a  e.  ( Prime  X.  NN ) ) )
6463impancom 441 . . . . . . . . . . 11  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6564expimpd 606 . . . . . . . . . 10  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6665exlimdvv 1773 . . . . . . . . 9  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6758, 66syl5bi 220 . . . . . . . 8  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6867ssrdv 3408 . . . . . . . . 9  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  C_  ( Prime  X.  NN ) )
6968sseld 3401 . . . . . . . 8  |-  ( ph  ->  ( b  e.  U_ p  e.  P  ( { p }  X.  K )  ->  b  e.  ( Prime  X.  NN ) ) )
7067, 69anim12d 565 . . . . . . 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 6783 . . . . . . . . . . 11  |-  ( a  e.  ( Prime  X.  NN )  ->  a  =  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7271fveq2d 5824 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. ) )
73 df-ov 6247 . . . . . . . . . 10  |-  ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7472, 73syl6eqr 2475 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ( 1st `  a
) ^ ( 2nd `  a ) ) )
75 1st2nd2 6783 . . . . . . . . . . 11  |-  ( b  e.  ( Prime  X.  NN )  ->  b  =  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7675fveq2d 5824 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. ) )
77 df-ov 6247 . . . . . . . . . 10  |-  ( ( 1st `  b ) ^ ( 2nd `  b
) )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7876, 77syl6eqr 2475 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) )
7974, 78eqeqan12d 2439 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  ( ( 1st `  a ) ^
( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) ) )
80 xp1st 6776 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 1st `  a
)  e.  Prime )
81 xp2nd 6777 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 2nd `  a
)  e.  NN )
8280, 81jca 534 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ( 1st `  a )  e.  Prime  /\  ( 2nd `  a
)  e.  NN ) )
83 xp1st 6776 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 1st `  b
)  e.  Prime )
84 xp2nd 6777 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 2nd `  b
)  e.  NN )
8583, 84jca 534 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ( 1st `  b )  e.  Prime  /\  ( 2nd `  b
)  e.  NN ) )
86 prmexpb 14608 . . . . . . . . . 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 833 . . . . . . . . 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 479 . . . . . . . 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 6785 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ( 1st `  a
)  =  ( 1st `  b )  /\  ( 2nd `  a )  =  ( 2nd `  b
) )  <->  a  =  b ) )
9079, 88, 893bitrd 282 . . . . . . 7  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  a  =  b ) )
9170, 90syl6 34 . . . . . 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 7558 . . . . 5  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V )
93 f1f1orn 5780 . . . . 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 5820 . . . . . 6  |-  ( a  =  z  ->  ( ^ `  a )  =  ( ^ `  z ) )
96 eqid 2423 . . . . . 6  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  =  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )
9795, 96, 1fvmpt 5903 . . . . 5  |-  ( z  e.  U_ p  e.  P  ( { p }  X.  K )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
9897adantl 467 . . . 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 5820 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( ^ `  <. p ,  k >. )
)
10099, 4syl6eqr 2475 . . . . . . . . . . . . . . 15  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( p ^ k ) )
101100eleq1d 2485 . . . . . . . . . . . . . 14  |-  ( a  =  <. p ,  k
>.  ->  ( ( ^ `  a )  e.  A  <->  ( p ^ k )  e.  A ) )
10237, 101syl5ibrcom 225 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  ( ^ `  a )  e.  A ) )
103102impancom 441 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  ( ^ `  a
)  e.  A ) )
104103expimpd 606 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
105104exlimdvv 1773 . . . . . . . . . 10  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
10658, 105syl5bi 220 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  A ) )
107106imp 430 . . . . . . . 8  |-  ( (
ph  /\  a  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ^ `  a
)  e.  A )
108107, 96fmptd 6000 . . . . . . 7  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A )
109 frn 5690 . . . . . . 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 3402 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  y  e.  A
)
11246nfel1 2578 . . . . . . 7  |-  F/ x [_ y  /  x ]_ B  e.  CC
11347eleq1d 2485 . . . . . . 7  |-  ( x  =  y  ->  ( B  e.  CC  <->  [_ y  /  x ]_ B  e.  CC ) )
114112, 113rspc 3114 . . . . . 6  |-  ( y  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  [_ y  /  x ]_ B  e.  CC )
)
11539, 114mpan9 471 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  CC )
116111, 115syldan 472 . . . 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 13727 . . 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 2469 . 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 3402 . . . 4  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  x  e.  A
)
120119, 38syldan 472 . . 3  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  e.  CC )
121 eldif 3384 . . . . 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 5042 . . . . . . . . . 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 2433 . . . . . . . . . . 11  |-  ( a  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  a
)  <->  x  =  (
p ^ k ) ) )
124123rexiunxp 4932 . . . . . . . . . 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 252 . . . . . . . . 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 462 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  =  ( p ^ k ) )
127 simplr 760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  e.  A )
128126, 127eqeltrrd 2502 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( p ^ k
)  e.  A )
12914rbaibd 918 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p ^ k )  e.  A )  ->  (
( p  e.  P  /\  k  e.  K
)  <->  ( p  e. 
Prime  /\  k  e.  NN ) ) )
130129adantlr 719 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  (
p ^ k )  e.  A )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
131128, 130syldan 472 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
132131pm5.32da 645 . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  P  /\  k  e.  K )
) )
134 ancom 451 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^ k ) )  <->  ( x  =  ( p ^ k
)  /\  ( p  e.  Prime  /\  k  e.  NN ) ) )
135132, 133, 1343bitr4g 291 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( p  e.  P  /\  k  e.  K )  /\  x  =  ( p ^
k ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^
k ) ) ) )
1361352exbidv 1764 . . . . . . . . . . 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 2885 . . . . . . . . . . 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 2885 . . . . . . . . . . 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 291 . . . . . . . . . 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 3402 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  NN )
142 isppw2 23979 . . . . . . . . . . 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 259 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^ k )  <->  (Λ `  x
)  =/=  0 ) )
145125, 144syl5bb 260 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) )  <->  (Λ `  x
)  =/=  0 ) )
146145necon2bbid 2639 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =  0  <->  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
147146pm5.32da 645 . . . . . 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 435 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  (Λ `  x
)  =  0 )  ->  B  =  0 ) )
150147, 149sylbird 238 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
151121, 150syl5bi 220 . . . 4  |-  ( ph  ->  ( x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
152151imp 430 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) ) )  ->  B  = 
0 )
153110, 120, 152, 12fsumss 13729 . 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 2464 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 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2594   A.wral 2709   E.wrex 2710   _Vcvv 3017   [_csb 3333    \ cdif 3371    C_ wss 3374   {csn 3936   <.cop 3942   U_ciun 4237    |-> cmpt 4420    X. cxp 4789   ran crn 4792   -->wf 5535   -1-1->wf1 5536   -1-1-onto->wf1o 5538   ` cfv 5539  (class class class)co 6244   1stc1st 6744   2ndc2nd 6745   Fincfn 7519   CCcc 9483   0cc0 9485   NNcn 10555   ^cexp 12217   sum_csu 13690   Primecprime 14560  Λcvma 23955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-rep 4474  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-inf2 8094  ax-cnex 9541  ax-resscn 9542  ax-1cn 9543  ax-icn 9544  ax-addcl 9545  ax-addrcl 9546  ax-mulcl 9547  ax-mulrcl 9548  ax-mulcom 9549  ax-addass 9550  ax-mulass 9551  ax-distr 9552  ax-i2m1 9553  ax-1ne0 9554  ax-1rid 9555  ax-rnegex 9556  ax-rrecex 9557  ax-cnre 9558  ax-pre-lttri 9559  ax-pre-lttrn 9560  ax-pre-ltadd 9561  ax-pre-mulgt0 9562  ax-pre-sup 9563  ax-addf 9564  ax-mulf 9565
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2275  df-mo 2276  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-ral 2714  df-rex 2715  df-reu 2716  df-rmo 2717  df-rab 2718  df-v 3019  df-sbc 3238  df-csb 3334  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-pss 3390  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4158  df-int 4194  df-iun 4239  df-iin 4240  df-br 4362  df-opab 4421  df-mpt 4422  df-tr 4457  df-eprel 4702  df-id 4706  df-po 4712  df-so 4713  df-fr 4750  df-se 4751  df-we 4752  df-xp 4797  df-rel 4798  df-cnv 4799  df-co 4800  df-dm 4801  df-rn 4802  df-res 4803  df-ima 4804  df-pred 5337  df-ord 5383  df-on 5384  df-lim 5385  df-suc 5386  df-iota 5503  df-fun 5541  df-fn 5542  df-f 5543  df-f1 5544  df-fo 5545  df-f1o 5546  df-fv 5547  df-isom 5548  df-riota 6206  df-ov 6247  df-oprab 6248  df-mpt2 6249  df-of 6484  df-om 6646  df-1st 6746  df-2nd 6747  df-supp 6865  df-wrecs 6978  df-recs 7040  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7473  df-en 7520  df-dom 7521  df-sdom 7522  df-fin 7523  df-fsupp 7832  df-fi 7873  df-sup 7904  df-inf 7905  df-oi 7973  df-card 8320  df-cda 8544  df-pnf 9623  df-mnf 9624  df-xr 9625  df-ltxr 9626  df-le 9627  df-sub 9808  df-neg 9809  df-div 10216  df-nn 10556  df-2 10614  df-3 10615  df-4 10616  df-5 10617  df-6 10618  df-7 10619  df-8 10620  df-9 10621  df-10 10622  df-n0 10816  df-z 10884  df-dec 10998  df-uz 11106  df-q 11211  df-rp 11249  df-xneg 11355  df-xadd 11356  df-xmul 11357  df-ioo 11585  df-ioc 11586  df-ico 11587  df-icc 11588  df-fz 11731  df-fzo 11862  df-fl 11973  df-mod 12042  df-seq 12159  df-exp 12218  df-fac 12405  df-bc 12433  df-hash 12461  df-shft 13069  df-cj 13101  df-re 13102  df-im 13103  df-sqrt 13237  df-abs 13238  df-limsup 13464  df-clim 13490  df-rlim 13491  df-sum 13691  df-ef 14059  df-sin 14061  df-cos 14062  df-pi 14064  df-dvds 14244  df-gcd 14407  df-prm 14561  df-pc 14725  df-struct 15061  df-ndx 15062  df-slot 15063  df-base 15064  df-sets 15065  df-ress 15066  df-plusg 15141  df-mulr 15142  df-starv 15143  df-sca 15144  df-vsca 15145  df-ip 15146  df-tset 15147  df-ple 15148  df-ds 15150  df-unif 15151  df-hom 15152  df-cco 15153  df-rest 15259  df-topn 15260  df-0g 15278  df-gsum 15279  df-topgen 15280  df-pt 15281  df-prds 15284  df-xrs 15338  df-qtop 15344  df-imas 15345  df-xps 15348  df-mre 15430  df-mrc 15431  df-acs 15433  df-mgm 16426  df-sgrp 16465  df-mnd 16475  df-submnd 16521  df-mulg 16614  df-cntz 16909  df-cmn 17370  df-psmet 18900  df-xmet 18901  df-met 18902  df-bl 18903  df-mopn 18904  df-fbas 18905  df-fg 18906  df-cnfld 18909  df-top 19858  df-bases 19859  df-topon 19860  df-topsp 19861  df-cld 19971  df-ntr 19972  df-cls 19973  df-nei 20051  df-lp 20089  df-perf 20090  df-cn 20180  df-cnp 20181  df-haus 20268  df-tx 20514  df-hmeo 20707  df-fil 20798  df-fm 20890  df-flim 20891  df-flf 20892  df-xms 21272  df-ms 21273  df-tms 21274  df-cncf 21847  df-limc 22758  df-dv 22759  df-log 23443  df-vma 23961
This theorem is referenced by:  fsumvma2  24079  vmasum  24081
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