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Theorem musum 20379
Description: The sum of the Möbius function over the divisors of  N gives one if  N  =  1, but otherwise always sums to zero. This makes the Möbius function useful for inverting divisor sums; see also muinv 20381. (Contributed by Mario Carneiro, 2-Jul-2015.)
Assertion
Ref Expression
musum  |-  ( N  e.  NN  ->  sum_ k  e.  { n  e.  NN  |  n  ||  N } 
( mmu `  k
)  =  if ( N  =  1 ,  1 ,  0 ) )
Distinct variable group:    k, n, N

Proof of Theorem musum
StepHypRef Expression
1 fveq2 5444 . . . . . . . 8  |-  ( n  =  k  ->  (
mmu `  n )  =  ( mmu `  k ) )
21neeq1d 2432 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =/=  0  <->  (
mmu `  k )  =/=  0 ) )
3 breq1 3986 . . . . . . 7  |-  ( n  =  k  ->  (
n  ||  N  <->  k  ||  N ) )
42, 3anbi12d 694 . . . . . 6  |-  ( n  =  k  ->  (
( ( mmu `  n )  =/=  0  /\  n  ||  N )  <-> 
( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
54elrab 2891 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  <->  ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
6 muval2 20320 . . . . . 6  |-  ( ( k  e.  NN  /\  ( mmu `  k )  =/=  0 )  -> 
( mmu `  k
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  k } ) ) )
76adantrr 700 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
85, 7sylbi 189 . . . 4  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ->  ( mmu `  k )  =  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  k } ) ) )
98adantl 454 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
109sumeq2dv 12127 . 2  |-  ( N  e.  NN  ->  sum_ k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ( mmu `  k )  =  sum_ k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  k } ) ) )
11 simpr 449 . . . . 5  |-  ( ( ( mmu `  n
)  =/=  0  /\  n  ||  N )  ->  n  ||  N
)
1211a1i 12 . . . 4  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  n  ||  N
) )
1312ss2rabdv 3215 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  { n  e.  NN  |  n  ||  N } )
14 ssrab2 3219 . . . . . 6  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  NN
15 simpr 449 . . . . . 6  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  {
n  e.  NN  | 
( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )
1614, 15sseldi 3139 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  NN )
17 mucl 20327 . . . . 5  |-  ( k  e.  NN  ->  (
mmu `  k )  e.  ZZ )
1816, 17syl 17 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  ZZ )
1918zcnd 10071 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  CC )
20 difrab 3403 . . . . . . 7  |-  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  =  {
n  e.  NN  | 
( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) }
21 pm3.21 437 . . . . . . . . . . 11  |-  ( n 
||  N  ->  (
( mmu `  n
)  =/=  0  -> 
( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) )
2221necon1bd 2487 . . . . . . . . . 10  |-  ( n 
||  N  ->  ( -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  ( mmu `  n )  =  0 ) )
2322imp 420 . . . . . . . . 9  |-  ( ( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 )
2423a1i 12 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 ) )
2524ss2rabi 3216 . . . . . . 7  |-  { n  e.  NN  |  ( n 
||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) }  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2620, 25eqsstri 3169 . . . . . 6  |-  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2726sseli 3137 . . . . 5  |-  ( k  e.  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  {
n  e.  NN  | 
( mmu `  n
)  =  0 } )
281eqeq1d 2264 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =  0  <->  (
mmu `  k )  =  0 ) )
2928elrab 2891 . . . . . 6  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  <->  ( k  e.  NN  /\  ( mmu `  k )  =  0 ) )
3029simprbi 452 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  ->  ( mmu `  k )  =  0 )
3127, 30syl 17 . . . 4  |-  ( k  e.  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  0 )
3231adantl 454 . . 3  |-  ( ( N  e.  NN  /\  k  e.  ( {
n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } ) )  -> 
( mmu `  k
)  =  0 )
33 fzfid 10987 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
34 sgmss 20292 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N ) )
35 ssfi 7037 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N
) )  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3633, 34, 35syl2anc 645 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3713, 19, 32, 36fsumss 12149 . 2  |-  ( N  e.  NN  ->  sum_ k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ( mmu `  k )  =  sum_ k  e.  { n  e.  NN  |  n  ||  N }  ( mmu `  k ) )
38 fveq2 5444 . . . . 5  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( # `  x
)  =  ( # `  { p  e.  Prime  |  p  ||  k } ) )
3938oveq2d 5794 . . . 4  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( -u 1 ^ ( # `  x
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
40 ssfi 7037 . . . . 5  |-  ( ( { n  e.  NN  |  n  ||  N }  e.  Fin  /\  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  { n  e.  NN  |  n  ||  N } )  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  e.  Fin )
4136, 13, 40syl2anc 645 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  e.  Fin )
42 eqid 2256 . . . . 5  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  =  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }
43 eqid 2256 . . . . 5  |-  ( m  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  |->  { p  e.  Prime  |  p 
||  m } )  =  ( m  e. 
{ n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  |->  { p  e. 
Prime  |  p  ||  m } )
44 oveq1 5785 . . . . . . . 8  |-  ( q  =  p  ->  (
q  pCnt  x )  =  ( p  pCnt  x ) )
4544cbvmptv 4071 . . . . . . 7  |-  ( q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  x ) )
46 oveq2 5786 . . . . . . . 8  |-  ( x  =  m  ->  (
p  pCnt  x )  =  ( p  pCnt  m ) )
4746mpteq2dv 4067 . . . . . . 7  |-  ( x  =  m  ->  (
p  e.  Prime  |->  ( p 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4845, 47syl5eq 2300 . . . . . 6  |-  ( x  =  m  ->  (
q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4948cbvmptv 4071 . . . . 5  |-  ( x  e.  NN  |->  ( q  e.  Prime  |->  ( q 
pCnt  x ) ) )  =  ( m  e.  NN  |->  ( p  e. 
Prime  |->  ( p  pCnt  m ) ) )
5042, 43, 49sqff1o 20368 . . . 4  |-  ( N  e.  NN  ->  (
m  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  |->  { p  e.  Prime  |  p 
||  m } ) : { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } -1-1-onto-> ~P { p  e. 
Prime  |  p  ||  N } )
51 breq2 3987 . . . . . . 7  |-  ( m  =  k  ->  (
p  ||  m  <->  p  ||  k
) )
5251rabbidv 2749 . . . . . 6  |-  ( m  =  k  ->  { p  e.  Prime  |  p  ||  m }  =  {
p  e.  Prime  |  p 
||  k } )
53 zex 9986 . . . . . . . 8  |-  ZZ  e.  _V
54 prmz 12709 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ZZ )
5554ssriv 3145 . . . . . . . 8  |-  Prime  C_  ZZ
5653, 55ssexi 4119 . . . . . . 7  |-  Prime  e.  _V
5756rabex 4125 . . . . . 6  |-  { p  e.  Prime  |  p  ||  k }  e.  _V
5852, 43, 57fvmpt 5522 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ->  ( ( m  e.  {
n  e.  NN  | 
( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  |->  { p  e. 
Prime  |  p  ||  m } ) `  k
)  =  { p  e.  Prime  |  p  ||  k } )
5958adantl 454 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( ( m  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  |->  { p  e.  Prime  |  p 
||  m } ) `
 k )  =  { p  e.  Prime  |  p  ||  k } )
60 neg1cn 9767 . . . . 5  |-  -u 1  e.  CC
61 prmdvdsfi 20293 . . . . . . 7  |-  ( N  e.  NN  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
62 elpwi 3593 . . . . . . 7  |-  ( x  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  x  C_  { p  e.  Prime  |  p 
||  N } )
63 ssfi 7037 . . . . . . 7  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  x  C_  { p  e.  Prime  |  p 
||  N } )  ->  x  e.  Fin )
6461, 62, 63syl2an 465 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  x  e.  Fin )
65 hashcl 11302 . . . . . 6  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
6664, 65syl 17 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 x )  e. 
NN0 )
67 expcl 11073 . . . . 5  |-  ( (
-u 1  e.  CC  /\  ( # `  x
)  e.  NN0 )  ->  ( -u 1 ^ ( # `  x
) )  e.  CC )
6860, 66, 67sylancr 647 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( -u 1 ^ ( # `  x ) )  e.  CC )
6939, 41, 50, 59, 68fsumf1o 12147 . . 3  |-  ( N  e.  NN  ->  sum_ x  e.  ~P  { p  e. 
Prime  |  p  ||  N }  ( -u 1 ^ ( # `  x
) )  =  sum_ k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ( -u 1 ^ ( # `  { p  e.  Prime  |  p  ||  k } ) ) )
70 fzfid 10987 . . . . 5  |-  ( N  e.  NN  ->  (
0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  e.  Fin )
7161adantr 453 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
72 pwfi 7105 . . . . . . 7  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin 
<->  ~P { p  e. 
Prime  |  p  ||  N }  e.  Fin )
7371, 72sylib 190 . . . . . 6  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ~P { p  e.  Prime  |  p  ||  N }  e.  Fin )
74 ssrab2 3219 . . . . . 6  |-  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  C_  ~P { p  e.  Prime  |  p  ||  N }
75 ssfi 7037 . . . . . 6  |-  ( ( ~P { p  e. 
Prime  |  p  ||  N }  e.  Fin  /\  {
s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z }  C_  ~P { p  e.  Prime  |  p  ||  N }
)  ->  { s  e.  ~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  e.  Fin )
7673, 74, 75sylancl 646 . . . . 5  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  e.  Fin )
77 simprr 736 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } )
78 fveq2 5444 . . . . . . . . . . 11  |-  ( s  =  x  ->  ( # `
 s )  =  ( # `  x
) )
7978eqeq1d 2264 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( # `  s )  =  z  <->  ( # `  x
)  =  z ) )
8079elrab 2891 . . . . . . . . 9  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  <-> 
( x  e.  ~P { p  e.  Prime  |  p  ||  N }  /\  ( # `  x
)  =  z ) )
8180simprbi 452 . . . . . . . 8  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ->  ( # `  x
)  =  z )
8277, 81syl 17 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  ( # `
 x )  =  z )
8382ralrimivva 2608 . . . . . 6  |-  ( N  e.  NN  ->  A. z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) A. x  e.  { s  e.  ~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( # `  x
)  =  z )
84 invdisj 3972 . . . . . 6  |-  ( A. z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) A. x  e. 
{ s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( # `  x
)  =  z  -> Disj  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z } )
8583, 84syl 17 . . . . 5  |-  ( N  e.  NN  -> Disj  z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) { s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z } )
8661adantr 453 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
8774, 77sseldi 3139 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  x  e.  ~P { p  e. 
Prime  |  p  ||  N } )
8887, 62syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  x  C_ 
{ p  e.  Prime  |  p  ||  N }
)
8986, 88, 63syl2anc 645 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  x  e.  Fin )
9089, 65syl 17 . . . . . 6  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  ( # `
 x )  e. 
NN0 )
9160, 90, 67sylancr 647 . . . . 5  |-  ( ( N  e.  NN  /\  ( z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )  ->  ( -u 1 ^ ( # `  x ) )  e.  CC )
9270, 76, 85, 91fsumiun 12230 . . . 4  |-  ( N  e.  NN  ->  sum_ x  e.  U_  z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  ( -u
1 ^ ( # `  x ) )  = 
sum_ z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) sum_ x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ ( # `  x
) ) )
9361adantr 453 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
94 elpwi 3593 . . . . . . . . . . . . 13  |-  ( s  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  s  C_  { p  e.  Prime  |  p 
||  N } )
9594adantl 454 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  C_ 
{ p  e.  Prime  |  p  ||  N }
)
96 ssdomg 6861 . . . . . . . . . . . 12  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( s  C_  { p  e.  Prime  |  p 
||  N }  ->  s  ~<_  { p  e.  Prime  |  p  ||  N }
) )
9793, 95, 96sylc 58 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  ~<_  { p  e.  Prime  |  p  ||  N }
)
98 ssfi 7037 . . . . . . . . . . . . 13  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  s  C_  { p  e.  Prime  |  p 
||  N } )  ->  s  e.  Fin )
9961, 94, 98syl2an 465 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  e.  Fin )
100 hashdom 11313 . . . . . . . . . . . 12  |-  ( ( s  e.  Fin  /\  { p  e.  Prime  |  p 
||  N }  e.  Fin )  ->  ( (
# `  s )  <_  ( # `  {
p  e.  Prime  |  p 
||  N } )  <-> 
s  ~<_  { p  e. 
Prime  |  p  ||  N } ) )
10199, 93, 100syl2anc 645 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  (
( # `  s )  <_  ( # `  {
p  e.  Prime  |  p 
||  N } )  <-> 
s  ~<_  { p  e. 
Prime  |  p  ||  N } ) )
10297, 101mpbird 225 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  <_ 
( # `  { p  e.  Prime  |  p  ||  N } ) )
103 hashcl 11302 . . . . . . . . . . . . 13  |-  ( s  e.  Fin  ->  ( # `
 s )  e. 
NN0 )
10499, 103syl 17 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e. 
NN0 )
105 nn0uz 10215 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
106104, 105syl6eleq 2346 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( ZZ>= `  0 )
)
107 hashcl 11302 . . . . . . . . . . . . . 14  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN0 )
10861, 107syl 17 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
109108adantr 453 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
110109nn0zd 10068 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  ZZ )
111 elfz5 10742 . . . . . . . . . . 11  |-  ( ( ( # `  s
)  e.  ( ZZ>= ` 
0 )  /\  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  ZZ )  ->  ( ( # `  s )  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) )  <->  ( # `  s
)  <_  ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )
112106, 110, 111syl2anc 645 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  (
( # `  s )  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  <->  ( # `  s
)  <_  ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )
113102, 112mpbird 225 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) )
114 eqidd 2257 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  =  ( # `  s
) )
115 eqeq2 2265 . . . . . . . . . 10  |-  ( z  =  ( # `  s
)  ->  ( ( # `
 s )  =  z  <->  ( # `  s
)  =  ( # `  s ) ) )
116115rcla4ev 2852 . . . . . . . . 9  |-  ( ( ( # `  s
)  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  /\  ( # `  s )  =  (
# `  s )
)  ->  E. z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) (
# `  s )  =  z )
117113, 114, 116syl2anc 645 . . . . . . . 8  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  E. z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) (
# `  s )  =  z )
118117ralrimiva 2599 . . . . . . 7  |-  ( N  e.  NN  ->  A. s  e.  ~P  { p  e. 
Prime  |  p  ||  N } E. z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( # `  s )  =  z )
119 rabid2 2690 . . . . . . 7  |-  ( ~P { p  e.  Prime  |  p  ||  N }  =  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  E. z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( # `  s )  =  z }  <->  A. s  e.  ~P  { p  e.  Prime  |  p 
||  N } E. z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) ( # `  s
)  =  z )
120118, 119sylibr 205 . . . . . 6  |-  ( N  e.  NN  ->  ~P { p  e.  Prime  |  p  ||  N }  =  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  E. z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( # `  s )  =  z } )
121 iunrab 3909 . . . . . 6  |-  U_ z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) { s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z }  =  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  E. z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( # `  s )  =  z }
122120, 121syl6reqr 2307 . . . . 5  |-  ( N  e.  NN  ->  U_ z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) { s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z }  =  ~P { p  e.  Prime  |  p  ||  N }
)
123122sumeq1d 12125 . . . 4  |-  ( N  e.  NN  ->  sum_ x  e.  U_  z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  ( -u
1 ^ ( # `  x ) )  = 
sum_ x  e.  ~P  { p  e.  Prime  |  p 
||  N }  ( -u 1 ^ ( # `  x ) ) )
124 elfznn0 10774 . . . . . . . . . 10  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  NN0 )
125124adantl 454 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  z  e.  NN0 )
126 expcl 11073 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  z  e.  NN0 )  ->  ( -u 1 ^ z )  e.  CC )
12760, 125, 126sylancr 647 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ( -u 1 ^ z )  e.  CC )
128 fsumconst 12203 . . . . . . . 8  |-  ( ( { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  e.  Fin  /\  ( -u 1 ^ z )  e.  CC )  ->  sum_ x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  ( -u
1 ^ z )  =  ( ( # `  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } )  x.  ( -u
1 ^ z ) ) )
12976, 127, 128syl2anc 645 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  sum_ x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ z )  =  ( ( # `  {
s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z } )  x.  ( -u 1 ^ z ) ) )
13081adantl 454 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } )  ->  ( # `  x
)  =  z )
131130oveq2d 5794 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  /\  x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } )  ->  ( -u 1 ^ ( # `  x
) )  =  (
-u 1 ^ z
) )
132131sumeq2dv 12127 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  sum_ x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ ( # `  x
) )  =  sum_ x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ z ) )
133 elfzelz 10750 . . . . . . . . 9  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  ZZ )
134 hashbc 11342 . . . . . . . . 9  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  z  e.  ZZ )  ->  (
( # `  { p  e.  Prime  |  p  ||  N } )  _C  z
)  =  ( # `  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )
13561, 133, 134syl2an 465 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  (
( # `  { p  e.  Prime  |  p  ||  N } )  _C  z
)  =  ( # `  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z } ) )
136135oveq1d 5793 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  (
( ( # `  {
p  e.  Prime  |  p 
||  N } )  _C  z )  x.  ( -u 1 ^ z ) )  =  ( ( # `  {
s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z } )  x.  ( -u 1 ^ z ) ) )
137129, 132, 1363eqtr4d 2298 . . . . . 6  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  sum_ x  e.  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ ( # `  x
) )  =  ( ( ( # `  {
p  e.  Prime  |  p 
||  N } )  _C  z )  x.  ( -u 1 ^ z ) ) )
138137sumeq2dv 12127 . . . . 5  |-  ( N  e.  NN  ->  sum_ z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) sum_ x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ ( # `  x
) )  =  sum_ z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) ( ( (
# `  { p  e.  Prime  |  p  ||  N } )  _C  z
)  x.  ( -u
1 ^ z ) ) )
139 ax-1cn 8749 . . . . . . . 8  |-  1  e.  CC
140139negidi 9069 . . . . . . 7  |-  ( 1  +  -u 1 )  =  0
141140oveq1i 5788 . . . . . 6  |-  ( ( 1  +  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  ( 0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )
142 binom1p 12240 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN0 )  -> 
( ( 1  + 
-u 1 ) ^
( # `  { p  e.  Prime  |  p  ||  N } ) )  = 
sum_ z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( ( ( # `  {
p  e.  Prime  |  p 
||  N } )  _C  z )  x.  ( -u 1 ^ z ) ) )
14360, 108, 142sylancr 647 . . . . . 6  |-  ( N  e.  NN  ->  (
( 1  +  -u
1 ) ^ ( # `
 { p  e. 
Prime  |  p  ||  N } ) )  = 
sum_ z  e.  ( 0 ... ( # `  { p  e.  Prime  |  p  ||  N }
) ) ( ( ( # `  {
p  e.  Prime  |  p 
||  N } )  _C  z )  x.  ( -u 1 ^ z ) ) )
144141, 143syl5eqr 2302 . . . . 5  |-  ( N  e.  NN  ->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  sum_ z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) ( ( (
# `  { p  e.  Prime  |  p  ||  N } )  _C  z
)  x.  ( -u
1 ^ z ) ) )
145 eqeq2 2265 . . . . . 6  |-  ( 1  =  if ( N  =  1 ,  1 ,  0 )  -> 
( ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  1  <->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) ) )
146 eqeq2 2265 . . . . . 6  |-  ( 0  =  if ( N  =  1 ,  1 ,  0 )  -> 
( ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  0  <->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) ) )
147 nprmdvds1 12738 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
148 simpr 449 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  N  =  1 )
149148breq2d 3995 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  ||  N 
<->  p  ||  1 ) )
150149notbid 287 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( -.  p  ||  N  <->  -.  p  ||  1
) )
151147, 150syl5ibr 214 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  e. 
Prime  ->  -.  p  ||  N
) )
152151ralrimiv 2598 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  A. p  e.  Prime  -.  p  ||  N )
153 rabeq0 3437 . . . . . . . . . . 11  |-  ( { p  e.  Prime  |  p 
||  N }  =  (/)  <->  A. p  e.  Prime  -.  p  ||  N )
154152, 153sylibr 205 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  { p  e. 
Prime  |  p  ||  N }  =  (/) )
155154fveq2d 5448 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  ( # `  (/) ) )
156 hash0 11307 . . . . . . . . 9  |-  ( # `  (/) )  =  0
157155, 156syl6eq 2304 . . . . . . . 8  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  0 )
158157oveq2d 5794 . . . . . . 7  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  ( 0 ^ 0 ) )
159 0cn 8785 . . . . . . . 8  |-  0  e.  CC
160 exp0 11060 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
161159, 160ax-mp 10 . . . . . . 7  |-  ( 0 ^ 0 )  =  1
162158, 161syl6eq 2304 . . . . . 6  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  1 )
163 df-ne 2421 . . . . . . . . . . 11  |-  ( N  =/=  1  <->  -.  N  =  1 )
164 eluz2b3 10244 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
165164biimpri 199 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =/=  1 )  ->  N  e.  ( ZZ>= ` 
2 ) )
166163, 165sylan2br 464 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  N  e.  ( ZZ>= `  2 )
)
167 exprmfct 12737 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
168166, 167syl 17 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  E. p  e.  Prime  p  ||  N
)
169 rabn0 3435 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  =/=  (/)  <->  E. p  e.  Prime  p  ||  N )
170168, 169sylibr 205 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  =/=  (/) )
17161adantr 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
172 hashnncl 11306 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( ( # `  { p  e.  Prime  |  p  ||  N }
)  e.  NN  <->  { p  e.  Prime  |  p  ||  N }  =/=  (/) ) )
173171, 172syl 17 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN  <->  { p  e.  Prime  |  p 
||  N }  =/=  (/) ) )
174170, 173mpbird 225 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN )
1751740expd 11213 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  0 )
176145, 146, 162, 175ifbothda 3555 . . . . 5  |-  ( N  e.  NN  ->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) )
177138, 144, 1763eqtr2d 2294 . . . 4  |-  ( N  e.  NN  ->  sum_ z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) sum_ x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ( -u 1 ^ ( # `  x
) )  =  if ( N  =  1 ,  1 ,  0 ) )
17892, 123, 1773eqtr3d 2296 . . 3  |-  ( N  e.  NN  ->  sum_ x  e.  ~P  { p  e. 
Prime  |  p  ||  N }  ( -u 1 ^ ( # `  x
) )  =  if ( N  =  1 ,  1 ,  0 ) )
17969, 178eqtr3d 2290 . 2  |-  ( N  e.  NN  ->  sum_ k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  k } ) )  =  if ( N  =  1 ,  1 ,  0 ) )
18010, 37, 1793eqtr3d 2296 1  |-  ( N  e.  NN  ->  sum_ k  e.  { n  e.  NN  |  n  ||  N } 
( mmu `  k
)  =  if ( N  =  1 ,  1 ,  0 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517   {crab 2520    \ cdif 3110    C_ wss 3113   (/)c0 3416   ifcif 3525   ~Pcpw 3585   U_ciun 3865  Disj wdisj 3953   class class class wbr 3983    e. cmpt 4037   ` cfv 4659  (class class class)co 5778    ~<_ cdom 6815   Fincfn 6817   CCcc 8689   0cc0 8691   1c1 8692    + caddc 8694    x. cmul 8696    <_ cle 8822   -ucneg 8992   NNcn 9700   2c2 9749   NN0cn0 9918   ZZcz 9977   ZZ>=cuz 10183   ...cfz 10734   ^cexp 11056    _C cbc 11267   #chash 11289   sum_csu 12109    || cdivides 12479   Primecprime 12706    pCnt cpc 12837   mmucmu 20280
This theorem is referenced by:  musumsum  20380  muinv  20381
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-disj 3954  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-2o 6434  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-cda 7748  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-q 10270  df-rp 10308  df-fz 10735  df-fzo 10823  df-fl 10877  df-mod 10926  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-clim 11913  df-sum 12110  df-divides 12480  df-gcd 12634  df-prime 12707  df-pc 12838  df-mu 20286
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