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Theorem musum 20447
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 20449. (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 5541 . . . . . . . 8  |-  ( n  =  k  ->  (
mmu `  n )  =  ( mmu `  k ) )
21neeq1d 2472 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =/=  0  <->  (
mmu `  k )  =/=  0 ) )
3 breq1 4042 . . . . . . 7  |-  ( n  =  k  ->  (
n  ||  N  <->  k  ||  N ) )
42, 3anbi12d 691 . . . . . 6  |-  ( n  =  k  ->  (
( ( mmu `  n )  =/=  0  /\  n  ||  N )  <-> 
( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
54elrab 2936 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  <->  ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
6 muval2 20388 . . . . . 6  |-  ( ( k  e.  NN  /\  ( mmu `  k )  =/=  0 )  -> 
( mmu `  k
)  =  ( -u
1 ^ ( # `  { p  e.  Prime  |  p  ||  k } ) ) )
76adantrr 697 . . . . 5  |-  ( ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
85, 7sylbi 187 . . . 4  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  ->  ( mmu `  k )  =  ( -u 1 ^ ( # `  {
p  e.  Prime  |  p 
||  k } ) ) )
98adantl 452 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
109sumeq2dv 12192 . 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 447 . . . . 5  |-  ( ( ( mmu `  n
)  =/=  0  /\  n  ||  N )  ->  n  ||  N
)
1211a1i 10 . . . 4  |-  ( ( N  e.  NN  /\  n  e.  NN )  ->  ( ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  n  ||  N
) )
1312ss2rabdv 3267 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  { n  e.  NN  |  n  ||  N } )
14 ssrab2 3271 . . . . . 6  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  NN
15 simpr 447 . . . . . 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 3191 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  NN )
17 mucl 20395 . . . . 5  |-  ( k  e.  NN  ->  (
mmu `  k )  e.  ZZ )
1816, 17syl 15 . . . 4  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  ZZ )
1918zcnd 10134 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  CC )
20 difrab 3455 . . . . . . 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 pm3.21 435 . . . . . . . . . . 11  |-  ( n 
||  N  ->  (
( mmu `  n
)  =/=  0  -> 
( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) )
2221necon1bd 2527 . . . . . . . . . 10  |-  ( n 
||  N  ->  ( -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N )  ->  ( mmu `  n )  =  0 ) )
2322imp 418 . . . . . . . . 9  |-  ( ( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 )
2423a1i 10 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( n  ||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) )  ->  ( mmu `  n )  =  0 ) )
2524ss2rabi 3268 . . . . . . 7  |-  { n  e.  NN  |  ( n 
||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) }  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2620, 25eqsstri 3221 . . . . . 6  |-  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2726sseli 3189 . . . . 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 2304 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =  0  <->  (
mmu `  k )  =  0 ) )
2928elrab 2936 . . . . . 6  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  <->  ( k  e.  NN  /\  ( mmu `  k )  =  0 ) )
3029simprbi 450 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( mmu `  n )  =  0 }  ->  ( mmu `  k )  =  0 )
3127, 30syl 15 . . . 4  |-  ( k  e.  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  =  0 )
3231adantl 452 . . 3  |-  ( ( N  e.  NN  /\  k  e.  ( {
n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } ) )  -> 
( mmu `  k
)  =  0 )
33 fzfid 11051 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
34 sgmss 20360 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N ) )
35 ssfi 7099 . . . 4  |-  ( ( ( 1 ... N
)  e.  Fin  /\  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N
) )  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3633, 34, 35syl2anc 642 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  e.  Fin )
3713, 19, 32, 36fsumss 12214 . 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 5541 . . . . 5  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( # `  x
)  =  ( # `  { p  e.  Prime  |  p  ||  k } ) )
3938oveq2d 5890 . . . 4  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( -u 1 ^ ( # `  x
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
40 ssfi 7099 . . . . 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 642 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  e.  Fin )
42 eqid 2296 . . . . 5  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  =  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }
43 eqid 2296 . . . . 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 5881 . . . . . . . 8  |-  ( q  =  p  ->  (
q  pCnt  x )  =  ( p  pCnt  x ) )
4544cbvmptv 4127 . . . . . . 7  |-  ( q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  x ) )
46 oveq2 5882 . . . . . . . 8  |-  ( x  =  m  ->  (
p  pCnt  x )  =  ( p  pCnt  m ) )
4746mpteq2dv 4123 . . . . . . 7  |-  ( x  =  m  ->  (
p  e.  Prime  |->  ( p 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4845, 47syl5eq 2340 . . . . . 6  |-  ( x  =  m  ->  (
q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4948cbvmptv 4127 . . . . 5  |-  ( x  e.  NN  |->  ( q  e.  Prime  |->  ( q 
pCnt  x ) ) )  =  ( m  e.  NN  |->  ( p  e. 
Prime  |->  ( p  pCnt  m ) ) )
5042, 43, 49sqff1o 20436 . . . 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 4043 . . . . . . 7  |-  ( m  =  k  ->  (
p  ||  m  <->  p  ||  k
) )
5251rabbidv 2793 . . . . . 6  |-  ( m  =  k  ->  { p  e.  Prime  |  p  ||  m }  =  {
p  e.  Prime  |  p 
||  k } )
53 zex 10049 . . . . . . . 8  |-  ZZ  e.  _V
54 prmz 12778 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ZZ )
5554ssriv 3197 . . . . . . . 8  |-  Prime  C_  ZZ
5653, 55ssexi 4175 . . . . . . 7  |-  Prime  e.  _V
5756rabex 4181 . . . . . 6  |-  { p  e.  Prime  |  p  ||  k }  e.  _V
5852, 43, 57fvmpt 5618 . . . . 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 452 . . . 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 9829 . . . . 5  |-  -u 1  e.  CC
61 prmdvdsfi 20361 . . . . . . 7  |-  ( N  e.  NN  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
62 elpwi 3646 . . . . . . 7  |-  ( x  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  x  C_  { p  e.  Prime  |  p 
||  N } )
63 ssfi 7099 . . . . . . 7  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  x  C_  { p  e.  Prime  |  p 
||  N } )  ->  x  e.  Fin )
6461, 62, 63syl2an 463 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  x  e.  Fin )
65 hashcl 11366 . . . . . 6  |-  ( x  e.  Fin  ->  ( # `
 x )  e. 
NN0 )
6664, 65syl 15 . . . . 5  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 x )  e. 
NN0 )
67 expcl 11137 . . . . 5  |-  ( (
-u 1  e.  CC  /\  ( # `  x
)  e.  NN0 )  ->  ( -u 1 ^ ( # `  x
) )  e.  CC )
6860, 66, 67sylancr 644 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( -u 1 ^ ( # `  x ) )  e.  CC )
6939, 41, 50, 59, 68fsumf1o 12212 . . 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 11051 . . . . 5  |-  ( N  e.  NN  ->  (
0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  e.  Fin )
7161adantr 451 . . . . . . 7  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
72 pwfi 7167 . . . . . . 7  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin 
<->  ~P { p  e. 
Prime  |  p  ||  N }  e.  Fin )
7371, 72sylib 188 . . . . . 6  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ~P { p  e.  Prime  |  p  ||  N }  e.  Fin )
74 ssrab2 3271 . . . . . 6  |-  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  C_  ~P { p  e.  Prime  |  p  ||  N }
75 ssfi 7099 . . . . . 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 643 . . . . 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 733 . . . . . . . 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 5541 . . . . . . . . . . 11  |-  ( s  =  x  ->  ( # `
 s )  =  ( # `  x
) )
7978eqeq1d 2304 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( # `  s )  =  z  <->  ( # `  x
)  =  z ) )
8079elrab 2936 . . . . . . . . 9  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  <-> 
( x  e.  ~P { p  e.  Prime  |  p  ||  N }  /\  ( # `  x
)  =  z ) )
8180simprbi 450 . . . . . . . 8  |-  ( x  e.  { s  e. 
~P { p  e. 
Prime  |  p  ||  N }  |  ( # `  s
)  =  z }  ->  ( # `  x
)  =  z )
8277, 81syl 15 . . . . . . 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 2648 . . . . . 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 4028 . . . . . 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 15 . . . . 5  |-  ( N  e.  NN  -> Disj  z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) { s  e.  ~P {
p  e.  Prime  |  p 
||  N }  | 
( # `  s )  =  z } )
8661adantr 451 . . . . . . . 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 3191 . . . . . . . . 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 15 . . . . . . . 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 642 . . . . . . 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 15 . . . . . 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 644 . . . . 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 12295 . . . 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 451 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
94 elpwi 3646 . . . . . . . . . . . . 13  |-  ( s  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  s  C_  { p  e.  Prime  |  p 
||  N } )
9594adantl 452 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  C_ 
{ p  e.  Prime  |  p  ||  N }
)
96 ssdomg 6923 . . . . . . . . . . . 12  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( s  C_  { p  e.  Prime  |  p 
||  N }  ->  s  ~<_  { p  e.  Prime  |  p  ||  N }
) )
9793, 95, 96sylc 56 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  ~<_  { p  e.  Prime  |  p  ||  N }
)
98 ssfi 7099 . . . . . . . . . . . . 13  |-  ( ( { p  e.  Prime  |  p  ||  N }  e.  Fin  /\  s  C_  { p  e.  Prime  |  p 
||  N } )  ->  s  e.  Fin )
9961, 94, 98syl2an 463 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  s  e.  Fin )
100 hashdom 11377 . . . . . . . . . . . 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 642 . . . . . . . . . . 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 223 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  <_ 
( # `  { p  e.  Prime  |  p  ||  N } ) )
103 hashcl 11366 . . . . . . . . . . . . 13  |-  ( s  e.  Fin  ->  ( # `
 s )  e. 
NN0 )
10499, 103syl 15 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e. 
NN0 )
105 nn0uz 10278 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
106104, 105syl6eleq 2386 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( ZZ>= `  0 )
)
107 hashcl 11366 . . . . . . . . . . . . . 14  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN0 )
10861, 107syl 15 . . . . . . . . . . . . 13  |-  ( N  e.  NN  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
109108adantr 451 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN0 )
110109nn0zd 10131 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  ZZ )
111 elfz5 10806 . . . . . . . . . . 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 642 . . . . . . . . . 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 223 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) )
114 eqidd 2297 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  =  ( # `  s
) )
115 eqeq2 2305 . . . . . . . . . 10  |-  ( z  =  ( # `  s
)  ->  ( ( # `
 s )  =  z  <->  ( # `  s
)  =  ( # `  s ) ) )
116115rspcev 2897 . . . . . . . . 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 642 . . . . . . . 8  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  E. z  e.  ( 0 ... ( # `
 { p  e. 
Prime  |  p  ||  N } ) ) (
# `  s )  =  z )
118117ralrimiva 2639 . . . . . . 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 2730 . . . . . . 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 203 . . . . . 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 3965 . . . . . 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 2347 . . . . 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 12190 . . . 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 10838 . . . . . . . . . 10  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  NN0 )
125124adantl 452 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  z  e.  NN0 )
126 expcl 11137 . . . . . . . . 9  |-  ( (
-u 1  e.  CC  /\  z  e.  NN0 )  ->  ( -u 1 ^ z )  e.  CC )
12760, 125, 126sylancr 644 . . . . . . . 8  |-  ( ( N  e.  NN  /\  z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) ) )  ->  ( -u 1 ^ z )  e.  CC )
128 fsumconst 12268 . . . . . . . 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 642 . . . . . . 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 452 . . . . . . . . 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 5890 . . . . . . . 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 12192 . . . . . . 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 10814 . . . . . . . . 9  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  ZZ )
134 hashbc 11407 . . . . . . . . 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 463 . . . . . . . 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 5889 . . . . . . 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 2338 . . . . . 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 12192 . . . . 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 8811 . . . . . . . 8  |-  1  e.  CC
140139negidi 9131 . . . . . . 7  |-  ( 1  +  -u 1 )  =  0
141140oveq1i 5884 . . . . . 6  |-  ( ( 1  +  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  ( 0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )
142 binom1p 12305 . . . . . . 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 644 . . . . . 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 2342 . . . . 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 2305 . . . . . 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 2305 . . . . . 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 12806 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
148 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  N  =  1 )
149148breq2d 4051 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  ||  N 
<->  p  ||  1 ) )
150149notbid 285 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( -.  p  ||  N  <->  -.  p  ||  1
) )
151147, 150syl5ibr 212 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( p  e. 
Prime  ->  -.  p  ||  N
) )
152151ralrimiv 2638 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  A. p  e.  Prime  -.  p  ||  N )
153 rabeq0 3489 . . . . . . . . . . 11  |-  ( { p  e.  Prime  |  p 
||  N }  =  (/)  <->  A. p  e.  Prime  -.  p  ||  N )
154152, 153sylibr 203 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  { p  e. 
Prime  |  p  ||  N }  =  (/) )
155154fveq2d 5545 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  ( # `  (/) ) )
156 hash0 11371 . . . . . . . . 9  |-  ( # `  (/) )  =  0
157155, 156syl6eq 2344 . . . . . . . 8  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  0 )
158157oveq2d 5890 . . . . . . 7  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  ( 0 ^ 0 ) )
159 0cn 8847 . . . . . . . 8  |-  0  e.  CC
160 exp0 11124 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
161159, 160ax-mp 8 . . . . . . 7  |-  ( 0 ^ 0 )  =  1
162158, 161syl6eq 2344 . . . . . 6  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  1 )
163 df-ne 2461 . . . . . . . . . . 11  |-  ( N  =/=  1  <->  -.  N  =  1 )
164 eluz2b3 10307 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  2
)  <->  ( N  e.  NN  /\  N  =/=  1 ) )
165164biimpri 197 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =/=  1 )  ->  N  e.  ( ZZ>= ` 
2 ) )
166163, 165sylan2br 462 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  N  e.  ( ZZ>= `  2 )
)
167 exprmfct 12805 . . . . . . . . . 10  |-  ( N  e.  ( ZZ>= `  2
)  ->  E. p  e.  Prime  p  ||  N
)
168166, 167syl 15 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  E. p  e.  Prime  p  ||  N
)
169 rabn0 3487 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  =/=  (/)  <->  E. p  e.  Prime  p  ||  N )
170168, 169sylibr 203 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  =/=  (/) )
17161adantr 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
172 hashnncl 11370 . . . . . . . . 9  |-  ( { p  e.  Prime  |  p 
||  N }  e.  Fin  ->  ( ( # `  { p  e.  Prime  |  p  ||  N }
)  e.  NN  <->  { p  e.  Prime  |  p  ||  N }  =/=  (/) ) )
173171, 172syl 15 . . . . . . . 8  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  NN  <->  { p  e.  Prime  |  p 
||  N }  =/=  (/) ) )
174170, 173mpbird 223 . . . . . . 7  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  e.  NN )
1751740expd 11277 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  0 )
176145, 146, 162, 175ifbothda 3608 . . . . 5  |-  ( N  e.  NN  ->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) )
177138, 144, 1763eqtr2d 2334 . . . 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 2336 . . 3  |-  ( N  e.  NN  ->  sum_ x  e.  ~P  { p  e. 
Prime  |  p  ||  N }  ( -u 1 ^ ( # `  x
) )  =  if ( N  =  1 ,  1 ,  0 ) )
17969, 178eqtr3d 2330 . 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 2336 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   U_ciun 3921  Disj wdisj 4009   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    ~<_ cdom 6877   Fincfn 6879   CCcc 8751   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    <_ cle 8884   -ucneg 9054   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798   ^cexp 11120    _C cbc 11331   #chash 11353   sum_csu 12174    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   mmucmu 20348
This theorem is referenced by:  musumsum  20448  muinv  20449
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
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-disj 4010  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-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-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  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-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  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-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-mu 20354
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