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Theorem fsumvma 20468
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 5555 . . . . 5  |-  ( ^ `  z )  e.  _V
21a1i 10 . . . 4  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  e.  _V )
3 fveq2 5541 . . . . . . . 8  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( ^ `  <. p ,  k >. )
)
4 df-ov 5877 . . . . . . . 8  |-  ( p ^ k )  =  ( ^ `  <. p ,  k >. )
53, 4syl6eqr 2346 . . . . . . 7  |-  ( z  =  <. p ,  k
>.  ->  ( ^ `  z )  =  ( p ^ k ) )
65eqeq2d 2307 . . . . . 6  |-  ( z  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  z
)  <->  x  =  (
p ^ k ) ) )
76biimpa 470 . . . . 5  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  x  =  ( p ^ k ) )
8 fsumvma.1 . . . . 5  |-  ( x  =  ( p ^
k )  ->  B  =  C )
97, 8syl 15 . . . 4  |-  ( ( z  =  <. p ,  k >.  /\  x  =  ( ^ `  z ) )  ->  B  =  C )
102, 9csbied 3136 . . 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 451 . . . 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 198 . . . . . . . 8  |-  ( ph  ->  ( ( p  e.  P  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) ) )
1615impl 603 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
( p  e.  Prime  /\  k  e.  NN )  /\  ( p ^
k )  e.  A
) )
1716simprd 449 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p ^ k )  e.  A )
1817ex 423 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
( p ^ k
)  e.  A ) )
1916simpld 445 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  (
p  e.  Prime  /\  k  e.  NN ) )
2019simpld 445 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  p  e.  Prime )
2120adantrr 697 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  p  e.  Prime )
2219simprd 449 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  k  e.  K )  ->  k  e.  NN )
2322adantrr 697 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  k  e.  NN )
2422ex 423 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  -> 
k  e.  NN ) )
2524ssrdv 3198 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  P )  ->  K  C_  NN )
2625sselda 3193 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  P )  /\  z  e.  K )  ->  z  e.  NN )
2726adantrl 696 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  z  e.  NN )
28 eqid 2296 . . . . . . . 8  |-  p  =  p
29 prmexpb 12812 . . . . . . . . 9  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  ( p  =  p  /\  k  =  z ) ) )
3029baibd 875 . . . . . . . 8  |-  ( ( ( ( p  e. 
Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  /\  p  =  p )  ->  ( ( p ^
k )  =  ( p ^ z )  <-> 
k  =  z ) )
3128, 30mpan2 652 . . . . . . 7  |-  ( ( ( p  e.  Prime  /\  p  e.  Prime )  /\  ( k  e.  NN  /\  z  e.  NN ) )  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3221, 21, 23, 27, 31syl22anc 1183 . . . . . 6  |-  ( ( ( ph  /\  p  e.  P )  /\  (
k  e.  K  /\  z  e.  K )
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) )
3332ex 423 . . . . 5  |-  ( (
ph  /\  p  e.  P )  ->  (
( k  e.  K  /\  z  e.  K
)  ->  ( (
p ^ k )  =  ( p ^
z )  <->  k  =  z ) ) )
3418, 33dom2lem 6917 . . . 4  |-  ( (
ph  /\  p  e.  P )  ->  (
k  e.  K  |->  ( p ^ k ) ) : K -1-1-> A
)
35 f1fi 7159 . . . 4  |-  ( ( A  e.  Fin  /\  ( k  e.  K  |->  ( p ^ k
) ) : K -1-1-> A )  ->  K  e.  Fin )
3613, 34, 35syl2anc 642 . . 3  |-  ( (
ph  /\  p  e.  P )  ->  K  e.  Fin )
3714simplbda 607 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p ^ k
)  e.  A )
38 fsumvma.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
3938ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. x  e.  A  B  e.  CC )
4039adantr 451 . . . 4  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  A. x  e.  A  B  e.  CC )
418eleq1d 2362 . . . . 5  |-  ( x  =  ( p ^
k )  ->  ( B  e.  CC  <->  C  e.  CC ) )
4241rspcv 2893 . . . 4  |-  ( ( p ^ k )  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  C  e.  CC ) )
4337, 40, 42sylc 56 . . 3  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  C  e.  CC )
4410, 11, 36, 43fsum2d 12250 . 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 2432 . . . 4  |-  F/_ y B
46 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ B
47 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4845, 46, 47cbvsumi 12186 . . 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 3097 . . . 4  |-  ( y  =  ( ^ `  z )  ->  [_ y  /  x ]_ B  = 
[_ ( ^ `  z )  /  x ]_ B )
50 snfi 6957 . . . . . . 7  |-  { p }  e.  Fin
51 xpfi 7144 . . . . . . 7  |-  ( ( { p }  e.  Fin  /\  K  e.  Fin )  ->  ( { p }  X.  K )  e. 
Fin )
5250, 36, 51sylancr 644 . . . . . 6  |-  ( (
ph  /\  p  e.  P )  ->  ( { p }  X.  K )  e.  Fin )
5352ralrimiva 2639 . . . . 5  |-  ( ph  ->  A. p  e.  P  ( { p }  X.  K )  e.  Fin )
54 iunfi 7160 . . . . 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 642 . . . 4  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  e.  Fin )
56 fvex 5555 . . . . . . 7  |-  ( ^ `  a )  e.  _V
5756a1ii 24 . . . . . 6  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  _V ) )
58 eliunxp 4839 . . . . . . . . 9  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  <->  E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) ) )
5914simprbda 606 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( p  e.  Prime  /\  k  e.  NN ) )
60 opelxp 4735 . . . . . . . . . . . . . 14  |-  ( <.
p ,  k >.  e.  ( Prime  X.  NN ) 
<->  ( p  e.  Prime  /\  k  e.  NN ) )
6159, 60sylibr 203 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  ->  <. p ,  k >.  e.  ( Prime  X.  NN ) )
62 eleq1 2356 . . . . . . . . . . . . 13  |-  ( a  =  <. p ,  k
>.  ->  ( a  e.  ( Prime  X.  NN ) 
<-> 
<. p ,  k >.  e.  ( Prime  X.  NN ) ) )
6361, 62syl5ibrcom 213 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  a  e.  ( Prime  X.  NN ) ) )
6463impancom 427 . . . . . . . . . . 11  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6564expimpd 586 . . . . . . . . . 10  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6665exlimdvv 1627 . . . . . . . . 9  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  a  e.  ( Prime  X.  NN ) ) )
6758, 66syl5bi 208 . . . . . . . 8  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  a  e.  ( Prime  X.  NN ) ) )
6867ssrdv 3198 . . . . . . . . 9  |-  ( ph  ->  U_ p  e.  P  ( { p }  X.  K )  C_  ( Prime  X.  NN ) )
6968sseld 3192 . . . . . . . 8  |-  ( ph  ->  ( b  e.  U_ p  e.  P  ( { p }  X.  K )  ->  b  e.  ( Prime  X.  NN ) ) )
7067, 69anim12d 546 . . . . . . 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 6175 . . . . . . . . . . 11  |-  ( a  e.  ( Prime  X.  NN )  ->  a  =  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7271fveq2d 5545 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. ) )
73 df-ov 5877 . . . . . . . . . 10  |-  ( ( 1st `  a ) ^ ( 2nd `  a
) )  =  ( ^ `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
7472, 73syl6eqr 2346 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ^ `  a )  =  ( ( 1st `  a
) ^ ( 2nd `  a ) ) )
75 1st2nd2 6175 . . . . . . . . . . 11  |-  ( b  e.  ( Prime  X.  NN )  ->  b  =  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7675fveq2d 5545 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. ) )
77 df-ov 5877 . . . . . . . . . 10  |-  ( ( 1st `  b ) ^ ( 2nd `  b
) )  =  ( ^ `  <. ( 1st `  b ) ,  ( 2nd `  b
) >. )
7876, 77syl6eqr 2346 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ^ `  b )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) )
7974, 78eqeqan12d 2311 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  ( ( 1st `  a ) ^
( 2nd `  a
) )  =  ( ( 1st `  b
) ^ ( 2nd `  b ) ) ) )
80 xp1st 6165 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 1st `  a
)  e.  Prime )
81 xp2nd 6166 . . . . . . . . . 10  |-  ( a  e.  ( Prime  X.  NN )  ->  ( 2nd `  a
)  e.  NN )
8280, 81jca 518 . . . . . . . . 9  |-  ( a  e.  ( Prime  X.  NN )  ->  ( ( 1st `  a )  e.  Prime  /\  ( 2nd `  a
)  e.  NN ) )
83 xp1st 6165 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 1st `  b
)  e.  Prime )
84 xp2nd 6166 . . . . . . . . . 10  |-  ( b  e.  ( Prime  X.  NN )  ->  ( 2nd `  b
)  e.  NN )
8583, 84jca 518 . . . . . . . . 9  |-  ( b  e.  ( Prime  X.  NN )  ->  ( ( 1st `  b )  e.  Prime  /\  ( 2nd `  b
)  e.  NN ) )
86 prmexpb 12812 . . . . . . . . . 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 799 . . . . . . . . 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 463 . . . . . . . 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 6177 . . . . . . . 8  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ( 1st `  a
)  =  ( 1st `  b )  /\  ( 2nd `  a )  =  ( 2nd `  b
) )  <->  a  =  b ) )
9079, 88, 893bitrd 270 . . . . . . 7  |-  ( ( a  e.  ( Prime  X.  NN )  /\  b  e.  ( Prime  X.  NN ) )  ->  (
( ^ `  a
)  =  ( ^ `  b )  <->  a  =  b ) )
9170, 90syl6 29 . . . . . 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 6917 . . . . 5  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) -1-1-> _V )
93 f1f1orn 5499 . . . . 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 15 . . . 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 5541 . . . . . 6  |-  ( a  =  z  ->  ( ^ `  a )  =  ( ^ `  z ) )
96 eqid 2296 . . . . . 6  |-  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  =  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )
9795, 96, 1fvmpt 5618 . . . . 5  |-  ( z  e.  U_ p  e.  P  ( { p }  X.  K )  -> 
( ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) `  z )  =  ( ^ `  z ) )
9897adantl 452 . . . 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 5541 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( ^ `  <. p ,  k >. )
)
10099, 4syl6eqr 2346 . . . . . . . . . . . . . . 15  |-  ( a  =  <. p ,  k
>.  ->  ( ^ `  a )  =  ( p ^ k ) )
101100eleq1d 2362 . . . . . . . . . . . . . 14  |-  ( a  =  <. p ,  k
>.  ->  ( ( ^ `  a )  e.  A  <->  ( p ^ k )  e.  A ) )
10237, 101syl5ibrcom 213 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( p  e.  P  /\  k  e.  K ) )  -> 
( a  =  <. p ,  k >.  ->  ( ^ `  a )  e.  A ) )
103102impancom 427 . . . . . . . . . . . 12  |-  ( (
ph  /\  a  =  <. p ,  k >.
)  ->  ( (
p  e.  P  /\  k  e.  K )  ->  ( ^ `  a
)  e.  A ) )
104103expimpd 586 . . . . . . . . . . 11  |-  ( ph  ->  ( ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
105104exlimdvv 1627 . . . . . . . . . 10  |-  ( ph  ->  ( E. p E. k ( a  = 
<. p ,  k >.  /\  ( p  e.  P  /\  k  e.  K
) )  ->  ( ^ `  a )  e.  A ) )
10658, 105syl5bi 208 . . . . . . . . 9  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  ->  ( ^ `  a )  e.  A ) )
107106imp 418 . . . . . . . 8  |-  ( (
ph  /\  a  e.  U_ p  e.  P  ( { p }  X.  K ) )  -> 
( ^ `  a
)  e.  A )
108107, 96fmptd 5700 . . . . . . 7  |-  ( ph  ->  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) :
U_ p  e.  P  ( { p }  X.  K ) --> A )
109 frn 5411 . . . . . . 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 15 . . . . . 6  |-  ( ph  ->  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) )  C_  A )
111110sselda 3193 . . . . 5  |-  ( (
ph  /\  y  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  y  e.  A
)
11246nfel1 2442 . . . . . . 7  |-  F/ x [_ y  /  x ]_ B  e.  CC
11347eleq1d 2362 . . . . . . 7  |-  ( x  =  y  ->  ( B  e.  CC  <->  [_ y  /  x ]_ B  e.  CC ) )
114112, 113rspc 2891 . . . . . 6  |-  ( y  e.  A  ->  ( A. x  e.  A  B  e.  CC  ->  [_ y  /  x ]_ B  e.  CC )
)
11539, 114mpan9 455 . . . . 5  |-  ( (
ph  /\  y  e.  A )  ->  [_ y  /  x ]_ B  e.  CC )
116111, 115syldan 456 . . . 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 12212 . . 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 2340 . 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 3193 . . . 4  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  x  e.  A
)
120119, 38syldan 456 . . 3  |-  ( (
ph  /\  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  e.  CC )
121 eldif 3175 . . . . 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 4946 . . . . . . . . . 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 2307 . . . . . . . . . . 11  |-  ( a  =  <. p ,  k
>.  ->  ( x  =  ( ^ `  a
)  <->  x  =  (
p ^ k ) ) )
124123rexiunxp 4842 . . . . . . . . . 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 240 . . . . . . . . 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 447 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  =  ( p ^ k ) )
127 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  ->  x  e.  A )
128126, 127eqeltrrd 2371 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( p ^ k
)  e.  A )
12914rbaibd 876 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( p ^ k )  e.  A )  ->  (
( p  e.  P  /\  k  e.  K
)  <->  ( p  e. 
Prime  /\  k  e.  NN ) ) )
130129adantlr 695 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  x  e.  A )  /\  (
p ^ k )  e.  A )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
131128, 130syldan 456 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  x  e.  A )  /\  x  =  ( p ^
k ) )  -> 
( ( p  e.  P  /\  k  e.  K )  <->  ( p  e.  Prime  /\  k  e.  NN ) ) )
132131pm5.32da 622 . . . . . . . . . . . . 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 437 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  P  /\  k  e.  K
)  /\  x  =  ( p ^ k
) )  <->  ( x  =  ( p ^
k )  /\  (
p  e.  P  /\  k  e.  K )
) )
134 ancom 437 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^ k ) )  <->  ( x  =  ( p ^ k
)  /\  ( p  e.  Prime  /\  k  e.  NN ) ) )
135132, 133, 1343bitr4g 279 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( p  e.  P  /\  k  e.  K )  /\  x  =  ( p ^
k ) )  <->  ( (
p  e.  Prime  /\  k  e.  NN )  /\  x  =  ( p ^
k ) ) ) )
1361352exbidv 1618 . . . . . . . . . . 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 2594 . . . . . . . . . . 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 2594 . . . . . . . . . . 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 279 . . . . . . . . . 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 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  NN )
142 isppw2 20369 . . . . . . . . . . 11  |-  ( x  e.  NN  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
143141, 142syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =/=  0  <->  E. p  e.  Prime  E. k  e.  NN  x  =  ( p ^
k ) ) )
144139, 143bitr4d 247 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( E. p  e.  P  E. k  e.  K  x  =  ( p ^ k )  <->  (Λ `  x
)  =/=  0 ) )
145125, 144syl5bb 248 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
x  e.  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) )  <->  (Λ `  x
)  =/=  0 ) )
146145necon2bbid 2517 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
(Λ `  x )  =  0  <->  -.  x  e.  ran  ( a  e.  U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) ) )
147146pm5.32da 622 . . . . . 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 423 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  (Λ `  x
)  =  0 )  ->  B  =  0 ) )
150147, 149sylbird 226 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  -.  x  e.  ran  ( a  e. 
U_ p  e.  P  ( { p }  X.  K )  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
151121, 150syl5bi 208 . . . 4  |-  ( ph  ->  ( x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) )  ->  B  =  0 ) )
152151imp 418 . . 3  |-  ( (
ph  /\  x  e.  ( A  \  ran  (
a  e.  U_ p  e.  P  ( {
p }  X.  K
)  |->  ( ^ `  a ) ) ) )  ->  B  = 
0 )
153110, 120, 152, 12fsumss 12214 . 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 2335 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801   [_csb 3094    \ cdif 3162    C_ wss 3165   {csn 3653   <.cop 3656   U_ciun 3921    e. cmpt 4093    X. cxp 4703   ran crn 4706   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Fincfn 6879   CCcc 8751   0cc0 8753   NNcn 9762   ^cexp 11120   sum_csu 12174   Primecprime 12774  Λcvma 20345
This theorem is referenced by:  fsumvma2  20469  vmasum  20471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-vma 20351
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