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Theorem musum 20976
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 20978. (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 5728 . . . . . . . 8  |-  ( n  =  k  ->  (
mmu `  n )  =  ( mmu `  k ) )
21neeq1d 2614 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =/=  0  <->  (
mmu `  k )  =/=  0 ) )
3 breq1 4215 . . . . . . 7  |-  ( n  =  k  ->  (
n  ||  N  <->  k  ||  N ) )
42, 3anbi12d 692 . . . . . 6  |-  ( n  =  k  ->  (
( ( mmu `  n )  =/=  0  /\  n  ||  N )  <-> 
( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
54elrab 3092 . . . . 5  |-  ( k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  <->  ( k  e.  NN  /\  ( ( mmu `  k )  =/=  0  /\  k  ||  N ) ) )
6 muval2 20917 . . . . . 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 12497 . 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 3424 . . 3  |-  ( N  e.  NN  ->  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  C_  { n  e.  NN  |  n  ||  N } )
14 ssrab2 3428 . . . . . 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 3346 . . . . 5  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  k  e.  NN )
17 mucl 20924 . . . . 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 10376 . . 3  |-  ( ( N  e.  NN  /\  k  e.  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  ->  ( mmu `  k )  e.  CC )
20 difrab 3615 . . . . . . 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 2672 . . . . . . . . . 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 3425 . . . . . . 7  |-  { n  e.  NN  |  ( n 
||  N  /\  -.  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) ) }  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2620, 25eqsstri 3378 . . . . . 6  |-  ( { n  e.  NN  |  n  ||  N }  \  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) } )  C_  { n  e.  NN  |  ( mmu `  n )  =  0 }
2726sseli 3344 . . . . 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 2444 . . . . . . 7  |-  ( n  =  k  ->  (
( mmu `  n
)  =  0  <->  (
mmu `  k )  =  0 ) )
2928elrab 3092 . . . . . 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 11312 . . . 4  |-  ( N  e.  NN  ->  (
1 ... N )  e. 
Fin )
34 sgmss 20889 . . . 4  |-  ( N  e.  NN  ->  { n  e.  NN  |  n  ||  N }  C_  ( 1 ... N ) )
35 ssfi 7329 . . . 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 12519 . 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 5728 . . . . 5  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( # `  x
)  =  ( # `  { p  e.  Prime  |  p  ||  k } ) )
3938oveq2d 6097 . . . 4  |-  ( x  =  { p  e. 
Prime  |  p  ||  k }  ->  ( -u 1 ^ ( # `  x
) )  =  (
-u 1 ^ ( # `
 { p  e. 
Prime  |  p  ||  k } ) ) )
40 ssfi 7329 . . . . 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 2436 . . . . 5  |-  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }  =  { n  e.  NN  |  ( ( mmu `  n )  =/=  0  /\  n  ||  N ) }
43 eqid 2436 . . . . 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 6088 . . . . . . . 8  |-  ( q  =  p  ->  (
q  pCnt  x )  =  ( p  pCnt  x ) )
4544cbvmptv 4300 . . . . . . 7  |-  ( q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  x ) )
46 oveq2 6089 . . . . . . . 8  |-  ( x  =  m  ->  (
p  pCnt  x )  =  ( p  pCnt  m ) )
4746mpteq2dv 4296 . . . . . . 7  |-  ( x  =  m  ->  (
p  e.  Prime  |->  ( p 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4845, 47syl5eq 2480 . . . . . 6  |-  ( x  =  m  ->  (
q  e.  Prime  |->  ( q 
pCnt  x ) )  =  ( p  e.  Prime  |->  ( p  pCnt  m ) ) )
4948cbvmptv 4300 . . . . 5  |-  ( x  e.  NN  |->  ( q  e.  Prime  |->  ( q 
pCnt  x ) ) )  =  ( m  e.  NN  |->  ( p  e. 
Prime  |->  ( p  pCnt  m ) ) )
5042, 43, 49sqff1o 20965 . . . 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 4216 . . . . . . 7  |-  ( m  =  k  ->  (
p  ||  m  <->  p  ||  k
) )
5251rabbidv 2948 . . . . . 6  |-  ( m  =  k  ->  { p  e.  Prime  |  p  ||  m }  =  {
p  e.  Prime  |  p 
||  k } )
53 zex 10291 . . . . . . . 8  |-  ZZ  e.  _V
54 prmz 13083 . . . . . . . . 9  |-  ( p  e.  Prime  ->  p  e.  ZZ )
5554ssriv 3352 . . . . . . . 8  |-  Prime  C_  ZZ
5653, 55ssexi 4348 . . . . . . 7  |-  Prime  e.  _V
5756rabex 4354 . . . . . 6  |-  { p  e.  Prime  |  p  ||  k }  e.  _V
5852, 43, 57fvmpt 5806 . . . . 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 10067 . . . . 5  |-  -u 1  e.  CC
61 prmdvdsfi 20890 . . . . . . 7  |-  ( N  e.  NN  ->  { p  e.  Prime  |  p  ||  N }  e.  Fin )
62 elpwi 3807 . . . . . . 7  |-  ( x  e.  ~P { p  e.  Prime  |  p  ||  N }  ->  x  C_  { p  e.  Prime  |  p 
||  N } )
63 ssfi 7329 . . . . . . 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 11639 . . . . . 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 11399 . . . . 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 12517 . . 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 11312 . . . . 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 7402 . . . . . . 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 3428 . . . . . 6  |-  { s  e.  ~P { p  e.  Prime  |  p  ||  N }  |  ( # `
 s )  =  z }  C_  ~P { p  e.  Prime  |  p  ||  N }
75 ssfi 7329 . . . . . 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 5728 . . . . . . . . . . 11  |-  ( s  =  x  ->  ( # `
 s )  =  ( # `  x
) )
7978eqeq1d 2444 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( # `  s )  =  z  <->  ( # `  x
)  =  z ) )
8079elrab 3092 . . . . . . . . 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 2798 . . . . . 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 4201 . . . . . 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 3346 . . . . . . . . 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 12600 . . . 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 3807 . . . . . . . . . . . . 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 7153 . . . . . . . . . . . 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 7329 . . . . . . . . . . . . 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 11653 . . . . . . . . . . . 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 11639 . . . . . . . . . . . . 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 10520 . . . . . . . . . . . 12  |-  NN0  =  ( ZZ>= `  0 )
106104, 105syl6eleq 2526 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  e.  ( ZZ>= `  0 )
)
107 hashcl 11639 . . . . . . . . . . . . . 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 10373 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 { p  e. 
Prime  |  p  ||  N } )  e.  ZZ )
111 elfz5 11051 . . . . . . . . . . 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 2437 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  s  e.  ~P { p  e.  Prime  |  p  ||  N } )  ->  ( # `
 s )  =  ( # `  s
) )
115 eqeq2 2445 . . . . . . . . . 10  |-  ( z  =  ( # `  s
)  ->  ( ( # `
 s )  =  z  <->  ( # `  s
)  =  ( # `  s ) ) )
116115rspcev 3052 . . . . . . . . 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 2789 . . . . . . 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 2885 . . . . . . 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 4138 . . . . . 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 2487 . . . . 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 12495 . . . 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 11083 . . . . . . . . . 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 11399 . . . . . . . . 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 12573 . . . . . . . 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 6097 . . . . . . . 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 12497 . . . . . . 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 11059 . . . . . . . . 9  |-  ( z  e.  ( 0 ... ( # `  {
p  e.  Prime  |  p 
||  N } ) )  ->  z  e.  ZZ )
134 hashbc 11702 . . . . . . . . 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 6096 . . . . . . 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 2478 . . . . . 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 12497 . . . . 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 9048 . . . . . . . 8  |-  1  e.  CC
140139negidi 9369 . . . . . . 7  |-  ( 1  +  -u 1 )  =  0
141140oveq1i 6091 . . . . . 6  |-  ( ( 1  +  -u 1
) ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  ( 0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )
142 binom1p 12610 . . . . . . 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 2482 . . . . 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 2445 . . . . . 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 2445 . . . . . 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 13111 . . . . . . . . . . . . 13  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
148 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  N  =  1 )
149148breq2d 4224 . . . . . . . . . . . . . 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 2788 . . . . . . . . . . 11  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  A. p  e.  Prime  -.  p  ||  N )
153 rabeq0 3649 . . . . . . . . . . 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 5732 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  ( # `  (/) ) )
156 hash0 11646 . . . . . . . . 9  |-  ( # `  (/) )  =  0
157155, 156syl6eq 2484 . . . . . . . 8  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( # `  {
p  e.  Prime  |  p 
||  N } )  =  0 )
158157oveq2d 6097 . . . . . . 7  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  ( 0 ^ 0 ) )
159 0cn 9084 . . . . . . . 8  |-  0  e.  CC
160 exp0 11386 . . . . . . . 8  |-  ( 0  e.  CC  ->  (
0 ^ 0 )  =  1 )
161159, 160ax-mp 8 . . . . . . 7  |-  ( 0 ^ 0 )  =  1
162158, 161syl6eq 2484 . . . . . 6  |-  ( ( N  e.  NN  /\  N  =  1 )  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  1 )
163 df-ne 2601 . . . . . . . . . . 11  |-  ( N  =/=  1  <->  -.  N  =  1 )
164 eluz2b3 10549 . . . . . . . . . . . 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 13110 . . . . . . . . . 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 3647 . . . . . . . . 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 11645 . . . . . . . . 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 11539 . . . . . 6  |-  ( ( N  e.  NN  /\  -.  N  =  1
)  ->  ( 0 ^ ( # `  {
p  e.  Prime  |  p 
||  N } ) )  =  0 )
176145, 146, 162, 175ifbothda 3769 . . . . 5  |-  ( N  e.  NN  ->  (
0 ^ ( # `  { p  e.  Prime  |  p  ||  N }
) )  =  if ( N  =  1 ,  1 ,  0 ) )
177138, 144, 1763eqtr2d 2474 . . . 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 2476 . . 3  |-  ( N  e.  NN  ->  sum_ x  e.  ~P  { p  e. 
Prime  |  p  ||  N }  ( -u 1 ^ ( # `  x
) )  =  if ( N  =  1 ,  1 ,  0 ) )
17969, 178eqtr3d 2470 . 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 2476 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 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3317    C_ wss 3320   (/)c0 3628   ifcif 3739   ~Pcpw 3799   U_ciun 4093  Disj wdisj 4182   class class class wbr 4212    e. cmpt 4266   ` cfv 5454  (class class class)co 6081    ~<_ cdom 7107   Fincfn 7109   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    <_ cle 9121   -ucneg 9292   NNcn 10000   2c2 10049   NN0cn0 10221   ZZcz 10282   ZZ>=cuz 10488   ...cfz 11043   ^cexp 11382    _C cbc 11593   #chash 11618   sum_csu 12479    || cdivides 12852   Primecprime 13079    pCnt cpc 13210   mmucmu 20877
This theorem is referenced by:  musumsum  20977  muinv  20978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-mu 20883
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