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Theorem musum 20844
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 20846. (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
Dummy variables  m  p  q  s  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5669 . . . . . . . 8  |-  ( n  =  k  ->  (
mmu `  n )  =  ( mmu `  k ) )
21neeq1d 2564 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =/=  0  <->  (
mmu `  k )  =/=  0 ) )
3 breq1 4157 . . . . . . 7  |-  ( n  =  k  ->  (
n  ||  N  <->  k  ||  N ) )
42, 3anbi12d 692 . . . . . 6  |-  ( n  =  k  ->  (
( ( mmu `  n )  =/=  0  /\  n  ||  N )  <-> 
( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
54elrab 3036 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  <->  ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
6 muval2 20785 . . . . . 6  |-  ( ( k  e.  NN  /\  ( mmu `  k )  =/=  0 )  -> 
( mmu `  k
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  k } ) ) )
76adantrr 698 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
85, 7sylbi 188 . . . 4  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ->  ( mmu `  k )  =  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  k } ) ) )
98adantl 453 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
109sumeq2dv 12425 . 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 448 . . . . 5  |-  ( ( ( mmu `  n
)  =/=  0  /\  n  ||  N )  ->  n  ||  N
)
1211a1i 11 . . . 4  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  n  ||  N
) )
1312ss2rabdv 3368 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  { n  e.  NN  |  n  ||  N } )
14 ssrab2 3372 . . . . . 6  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  NN
15 simpr 448 . . . . . 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 3290 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  NN )
17 mucl 20792 . . . . 5  |-  ( k  e.  NN  ->  (
mmu `  k )  e.  ZZ )
1816, 17syl 16 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  ZZ )
1918zcnd 10309 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  CC )
20 difrab 3559 . . . . . . 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 pm3.21 436 . . . . . . . . . . 11  |-  ( n 
||  N  ->  (
( mmu `  n
)  =/=  0  -> 
( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) )
2221necon1bd 2619 . . . . . . . . . 10  |-  ( n 
||  N  ->  ( -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  ( mmu `  n )  =  0 ) )
2322imp 419 . . . . . . . . 9  |-  ( ( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 )
2423a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 ) )
2524ss2rabi 3369 . . . . . . 7  |-  { n  e.  NN  |  ( n 
||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) }  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2620, 25eqsstri 3322 . . . . . 6  |-  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2726sseli 3288 . . . . 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 2396 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =  0  <->  (
mmu `  k )  =  0 ) )
2928elrab 3036 . . . . . 6  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  <->  ( k  e.  NN  /\  ( mmu `  k )  =  0 ) )
3029simprbi 451 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  ->  ( mmu `  k )  =  0 )
3127, 30syl 16 . . . 4  |-  ( k  e.  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  0 )
3231adantl 453 . . 3  |-  ( ( N  e.  NN  /\  k  e.  ( {
n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } ) )  -> 
( mmu `  k
)  =  0 )
33 fzfid 11240 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
34 sgmss 20757 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N ) )
35 ssfi 7266 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N
) )  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3633, 34, 35syl2anc 643 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3713, 19, 32, 36fsumss 12447 . 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 5669 . . . . 5  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( # `  x
)  =  ( # `  { p  e.  Prime  |  p  ||  k } ) )
3938oveq2d 6037 . . . 4  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( -u 1 ^ ( # `  x
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
40 ssfi 7266 . . . . 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 643 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  e.  Fin )
42 eqid 2388 . . . . 5  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  =  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }
43 eqid 2388 . . . . 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 6028 . . . . . . . 8  |-  ( q  =  p  ->  (
q  pCnt  x )  =  ( p  pCnt  x ) )
4544cbvmptv 4242 . . . . . . 7  |-  ( q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  x ) )
46 oveq2 6029 . . . . . . . 8  |-  ( x  =  m  ->  (
p  pCnt  x )  =  ( p  pCnt  m ) )
4746mpteq2dv 4238 . . . . . . 7  |-  ( x  =  m  ->  (
p  e.  Prime  |->  ( p 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4845, 47syl5eq 2432 . . . . . 6  |-  ( x  =  m  ->  (
q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4948cbvmptv 4242 . . . . 5  |-  ( x  e.  NN  |->  ( q  e.  Prime  |->  ( q 
pCnt  x ) ) )  =  ( m  e.  NN  |->  ( p  e. 
Prime  |->  ( p  pCnt  m ) ) )
5042, 43, 49sqff1o 20833 . . . 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 4158 . . . . . . 7  |-  ( m  =  k  ->  (
p  ||  m  <->  p  ||  k
) )
5251rabbidv 2892 . . . . . 6  |-  ( m  =  k  ->  { p  e.  Prime  |  p  ||  m }  =  {
p  e.  Prime  |  p 
||  k } )
53 zex 10224 . . . . . . . 8  |-  ZZ  e.  _V
54 prmz 13011 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ZZ )
5554ssriv 3296 . . . . . . . 8  |-  Prime  C_  ZZ
5653, 55ssexi 4290 . . . . . . 7  |-  Prime  e.  _V
5756rabex 4296 . . . . . 6  |-  { p  e.  Prime  |  p  ||  k }  e.  _V
5852, 43, 57fvmpt 5746 . . . . 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 453 . . . 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 10000 . . . . 5  |-  -u 1  e.  CC
61 prmdvdsfi 20758 . . . . . . 7  |-  ( N  e.  NN  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
62 elpwi 3751 . . . . . . 7  |-  ( x  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  x  C_  { p  e.  Prime  |  p 
||  N } )
63 ssfi 7266 . . . . . . 7  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  x  C_  { p  e.  Prime  |  p 
||  N } )  ->  x  e.  Fin )
6461, 62, 63syl2an 464 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  x  e.  Fin )
65 hashcl 11567 . . . . . 6  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
6664, 65syl 16 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 x )  e. 
NN0 )
67 expcl 11327 . . . . 5  |-  ( (
-u 1  e.  CC  /\  ( # `  x
)  e.  NN0 )  ->  ( -u 1 ^ ( # `  x
) )  e.  CC )
6860, 66, 67sylancr 645 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( -u 1 ^ ( # `  x ) )  e.  CC )
6939, 41, 50, 59, 68fsumf1o 12445 . . 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 11240 . . . . 5  |-  ( N  e.  NN  ->  (
0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  e.  Fin )
7161adantr 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
72 pwfi 7338 . . . . . . 7  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin 
<->  ~P { p  e. 
Prime  |  p  ||  N }  e.  Fin )
7371, 72sylib 189 . . . . . 6  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ~P { p  e.  Prime  |  p  ||  N }  e.  Fin )
74 ssrab2 3372 . . . . . 6  |-  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  C_  ~P { p  e.  Prime  |  p  ||  N }
75 ssfi 7266 . . . . . 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 644 . . . . 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 734 . . . . . . . 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 5669 . . . . . . . . . . 11  |-  ( s  =  x  ->  ( # `
 s )  =  ( # `  x
) )
7978eqeq1d 2396 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( # `  s )  =  z  <->  ( # `  x
)  =  z ) )
8079elrab 3036 . . . . . . . . 9  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  <-> 
( x  e.  ~P { p  e.  Prime  |  p  ||  N }  /\  ( # `  x
)  =  z ) )
8180simprbi 451 . . . . . . . 8  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ->  ( # `  x
)  =  z )
8277, 81syl 16 . . . . . . 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 2742 . . . . . 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 4143 . . . . . 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 16 . . . . 5  |-  ( N  e.  NN  -> Disj  z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) { s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z } )
8661adantr 452 . . . . . . . 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 3290 . . . . . . . . 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 16 . . . . . . . 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 643 . . . . . . 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 16 . . . . . 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 645 . . . . 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 12528 . . . 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 452 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
94 elpwi 3751 . . . . . . . . . . . . 13  |-  ( s  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  s  C_  { p  e.  Prime  |  p 
||  N } )
9594adantl 453 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  C_ 
{ p  e.  Prime  |  p  ||  N }
)
96 ssdomg 7090 . . . . . . . . . . . 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 7266 . . . . . . . . . . . . 13  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  s  C_  { p  e.  Prime  |  p 
||  N } )  ->  s  e.  Fin )
9961, 94, 98syl2an 464 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  e.  Fin )
100 hashdom 11581 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 224 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  <_ 
( # `  { p  e.  Prime  |  p  ||  N } ) )
103 hashcl 11567 . . . . . . . . . . . . 13  |-  ( s  e.  Fin  ->  ( # `
 s )  e. 
NN0 )
10499, 103syl 16 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e. 
NN0 )
105 nn0uz 10453 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
106104, 105syl6eleq 2478 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( ZZ>= `  0 )
)
107 hashcl 11567 . . . . . . . . . . . . . 14  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN0 )
10861, 107syl 16 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
109108adantr 452 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
110109nn0zd 10306 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  ZZ )
111 elfz5 10984 . . . . . . . . . . 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 643 . . . . . . . . . 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 224 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) )
114 eqidd 2389 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  =  ( # `  s
) )
115 eqeq2 2397 . . . . . . . . . 10  |-  ( z  =  ( # `  s
)  ->  ( ( # `
 s )  =  z  <->  ( # `  s
)  =  ( # `  s ) ) )
116115rspcev 2996 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  E. z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) (
# `  s )  =  z )
118117ralrimiva 2733 . . . . . . 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 2829 . . . . . . 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 204 . . . . . 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 4080 . . . . . 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 2439 . . . . 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 12423 . . . 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 11016 . . . . . . . . . 10  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  NN0 )
125124adantl 453 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  z  e.  NN0 )
126 expcl 11327 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  z  e.  NN0 )  ->  ( -u 1 ^ z )  e.  CC )
12760, 125, 126sylancr 645 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ( -u 1 ^ z )  e.  CC )
128 fsumconst 12501 . . . . . . . 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 643 . . . . . . 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 453 . . . . . . . . 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 6037 . . . . . . . 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 12425 . . . . . . 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 10992 . . . . . . . . 9  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  ZZ )
134 hashbc 11630 . . . . . . . . 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 464 . . . . . . . 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 6036 . . . . . . 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 2430 . . . . . 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 12425 . . . . 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 8982 . . . . . . . 8  |-  1  e.  CC
140139negidi 9302 . . . . . . 7  |-  ( 1  +  -u 1 )  =  0
141140oveq1i 6031 . . . . . 6  |-  ( ( 1  +  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  ( 0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )
142 binom1p 12538 . . . . . . 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 645 . . . . . 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 2434 . . . . 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 2397 . . . . . 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 2397 . . . . . 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 13039 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
148 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  N  =  1 )
149148breq2d 4166 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  ||  N 
<->  p  ||  1 ) )
150149notbid 286 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( -.  p  ||  N  <->  -.  p  ||  1
) )
151147, 150syl5ibr 213 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  e. 
Prime  ->  -.  p  ||  N
) )
152151ralrimiv 2732 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  A. p  e.  Prime  -.  p  ||  N )
153 rabeq0 3593 . . . . . . . . . . 11  |-  ( { p  e.  Prime  |  p 
||  N }  =  (/)  <->  A. p  e.  Prime  -.  p  ||  N )
154152, 153sylibr 204 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  { p  e. 
Prime  |  p  ||  N }  =  (/) )
155154fveq2d 5673 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  ( # `  (/) ) )
156 hash0 11574 . . . . . . . . 9  |-  ( # `  (/) )  =  0
157155, 156syl6eq 2436 . . . . . . . 8  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  0 )
158157oveq2d 6037 . . . . . . 7  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  ( 0 ^ 0 ) )
159 0cn 9018 . . . . . . . 8  |-  0  e.  CC
160 exp0 11314 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
161159, 160ax-mp 8 . . . . . . 7  |-  ( 0 ^ 0 )  =  1
162158, 161syl6eq 2436 . . . . . 6  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  1 )
163 df-ne 2553 . . . . . . . . . . 11  |-  ( N  =/=  1  <->  -.  N  =  1 )
164 eluz2b3 10482 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
165164biimpri 198 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =/=  1 )  ->  N  e.  ( ZZ>= ` 
2 ) )
166163, 165sylan2br 463 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  N  e.  ( ZZ>= `  2 )
)
167 exprmfct 13038 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
168166, 167syl 16 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  E. p  e.  Prime  p  ||  N
)
169 rabn0 3591 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  =/=  (/)  <->  E. p  e.  Prime  p  ||  N )
170168, 169sylibr 204 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  =/=  (/) )
17161adantr 452 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
172 hashnncl 11573 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( ( # `  { p  e.  Prime  |  p  ||  N }
)  e.  NN  <->  { p  e.  Prime  |  p  ||  N }  =/=  (/) ) )
173171, 172syl 16 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN  <->  { p  e.  Prime  |  p 
||  N }  =/=  (/) ) )
174170, 173mpbird 224 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN )
1751740expd 11467 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  0 )
176145, 146, 162, 175ifbothda 3713 . . . . 5  |-  ( N  e.  NN  ->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) )
177138, 144, 1763eqtr2d 2426 . . . 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 2428 . . 3  |-  ( N  e.  NN  ->  sum_ x  e.  ~P  { p  e. 
Prime  |  p  ||  N }  ( -u 1 ^ ( # `  x
) )  =  if ( N  =  1 ,  1 ,  0 ) )
17969, 178eqtr3d 2422 . 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 2428 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 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   A.wral 2650   E.wrex 2651   {crab 2654    \ cdif 3261    C_ wss 3264   (/)c0 3572   ifcif 3683   ~Pcpw 3743   U_ciun 4036  Disj wdisj 4124   class class class wbr 4154    e. cmpt 4208   ` cfv 5395  (class class class)co 6021    ~<_ cdom 7044   Fincfn 7046   CCcc 8922   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    <_ cle 9055   -ucneg 9225   NNcn 9933   2c2 9982   NN0cn0 10154   ZZcz 10215   ZZ>=cuz 10421   ...cfz 10976   ^cexp 11310    _C cbc 11521   #chash 11546   sum_csu 12407    || cdivides 12780   Primecprime 13007    pCnt cpc 13138   mmucmu 20745
This theorem is referenced by:  musumsum  20845  muinv  20846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-disj 4125  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-q 10508  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-sum 12408  df-dvds 12781  df-gcd 12935  df-prm 13008  df-pc 13139  df-mu 20751
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