Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( n = k -> ( mmu ` n ) = ( mmu ` k ) ) |
2 |
1
|
neeq1d |
|- ( n = k -> ( ( mmu ` n ) =/= 0 <-> ( mmu ` k ) =/= 0 ) ) |
3 |
|
breq1 |
|- ( n = k -> ( n || N <-> k || N ) ) |
4 |
2 3
|
anbi12d |
|- ( n = k -> ( ( ( mmu ` n ) =/= 0 /\ n || N ) <-> ( ( mmu ` k ) =/= 0 /\ k || N ) ) ) |
5 |
4
|
elrab |
|- ( k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } <-> ( k e. NN /\ ( ( mmu ` k ) =/= 0 /\ k || N ) ) ) |
6 |
|
muval2 |
|- ( ( k e. NN /\ ( mmu ` k ) =/= 0 ) -> ( mmu ` k ) = ( -u 1 ^ ( # ` { p e. Prime | p || k } ) ) ) |
7 |
6
|
adantrr |
|- ( ( k e. NN /\ ( ( mmu ` k ) =/= 0 /\ k || N ) ) -> ( mmu ` k ) = ( -u 1 ^ ( # ` { p e. Prime | p || k } ) ) ) |
8 |
5 7
|
sylbi |
|- ( k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } -> ( mmu ` k ) = ( -u 1 ^ ( # ` { p e. Prime | p || k } ) ) ) |
9 |
8
|
adantl |
|- ( ( N e. NN /\ k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> ( mmu ` k ) = ( -u 1 ^ ( # ` { p e. Prime | p || k } ) ) ) |
10 |
9
|
sumeq2dv |
|- ( 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 |
|- ( ( ( mmu ` n ) =/= 0 /\ n || N ) -> n || N ) |
12 |
11
|
a1i |
|- ( ( N e. NN /\ n e. NN ) -> ( ( ( mmu ` n ) =/= 0 /\ n || N ) -> n || N ) ) |
13 |
12
|
ss2rabdv |
|- ( N e. NN -> { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } C_ { n e. NN | n || N } ) |
14 |
|
ssrab2 |
|- { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } C_ NN |
15 |
|
simpr |
|- ( ( N e. NN /\ k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) |
16 |
14 15
|
sselid |
|- ( ( N e. NN /\ k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> k e. NN ) |
17 |
|
mucl |
|- ( k e. NN -> ( mmu ` k ) e. ZZ ) |
18 |
16 17
|
syl |
|- ( ( N e. NN /\ k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> ( mmu ` k ) e. ZZ ) |
19 |
18
|
zcnd |
|- ( ( N e. NN /\ k e. { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> ( mmu ` k ) e. CC ) |
20 |
|
difrab |
|- ( { 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 |
|- ( n || N -> ( ( mmu ` n ) =/= 0 -> ( ( mmu ` n ) =/= 0 /\ n || N ) ) ) |
22 |
21
|
necon1bd |
|- ( n || N -> ( -. ( ( mmu ` n ) =/= 0 /\ n || N ) -> ( mmu ` n ) = 0 ) ) |
23 |
22
|
imp |
|- ( ( n || N /\ -. ( ( mmu ` n ) =/= 0 /\ n || N ) ) -> ( mmu ` n ) = 0 ) |
24 |
23
|
a1i |
|- ( n e. NN -> ( ( n || N /\ -. ( ( mmu ` n ) =/= 0 /\ n || N ) ) -> ( mmu ` n ) = 0 ) ) |
25 |
24
|
ss2rabi |
|- { n e. NN | ( n || N /\ -. ( ( mmu ` n ) =/= 0 /\ n || N ) ) } C_ { n e. NN | ( mmu ` n ) = 0 } |
26 |
20 25
|
eqsstri |
|- ( { n e. NN | n || N } \ { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) C_ { n e. NN | ( mmu ` n ) = 0 } |
27 |
26
|
sseli |
|- ( k e. ( { n e. NN | n || N } \ { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> k e. { n e. NN | ( mmu ` n ) = 0 } ) |
28 |
|
fveqeq2 |
|- ( n = k -> ( ( mmu ` n ) = 0 <-> ( mmu ` k ) = 0 ) ) |
29 |
28
|
elrab |
|- ( k e. { n e. NN | ( mmu ` n ) = 0 } <-> ( k e. NN /\ ( mmu ` k ) = 0 ) ) |
30 |
29
|
simprbi |
|- ( k e. { n e. NN | ( mmu ` n ) = 0 } -> ( mmu ` k ) = 0 ) |
31 |
27 30
|
syl |
|- ( k e. ( { n e. NN | n || N } \ { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) -> ( mmu ` k ) = 0 ) |
32 |
31
|
adantl |
|- ( ( N e. NN /\ k e. ( { n e. NN | n || N } \ { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } ) ) -> ( mmu ` k ) = 0 ) |
33 |
|
fzfid |
|- ( N e. NN -> ( 1 ... N ) e. Fin ) |
34 |
|
dvdsssfz1 |
|- ( N e. NN -> { n e. NN | n || N } C_ ( 1 ... N ) ) |
35 |
33 34
|
ssfid |
|- ( N e. NN -> { n e. NN | n || N } e. Fin ) |
36 |
13 19 32 35
|
fsumss |
|- ( 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 ) ) |
37 |
|
fveq2 |
|- ( x = { p e. Prime | p || k } -> ( # ` x ) = ( # ` { p e. Prime | p || k } ) ) |
38 |
37
|
oveq2d |
|- ( x = { p e. Prime | p || k } -> ( -u 1 ^ ( # ` x ) ) = ( -u 1 ^ ( # ` { p e. Prime | p || k } ) ) ) |
39 |
35 13
|
ssfid |
|- ( N e. NN -> { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } e. Fin ) |
40 |
|
eqid |
|- { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } = { n e. NN | ( ( mmu ` n ) =/= 0 /\ n || N ) } |
41 |
|
eqid |
|- ( 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 } ) |
42 |
|
oveq1 |
|- ( q = p -> ( q pCnt x ) = ( p pCnt x ) ) |
43 |
42
|
cbvmptv |
|- ( q e. Prime |-> ( q pCnt x ) ) = ( p e. Prime |-> ( p pCnt x ) ) |
44 |
|
oveq2 |
|- ( x = m -> ( p pCnt x ) = ( p pCnt m ) ) |
45 |
44
|
mpteq2dv |
|- ( x = m -> ( p e. Prime |-> ( p pCnt x ) ) = ( p e. Prime |-> ( p pCnt m ) ) ) |
46 |
43 45
|
syl5eq |
|- ( x = m -> ( q e. Prime |-> ( q pCnt x ) ) = ( p e. Prime |-> ( p pCnt m ) ) ) |
47 |
46
|
cbvmptv |
|- ( x e. NN |-> ( q e. Prime |-> ( q pCnt x ) ) ) = ( m e. NN |-> ( p e. Prime |-> ( p pCnt m ) ) ) |
48 |
40 41 47
|
sqff1o |
|- ( 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 } ) |
49 |
|
breq2 |
|- ( m = k -> ( p || m <-> p || k ) ) |
50 |
49
|
rabbidv |
|- ( m = k -> { p e. Prime | p || m } = { p e. Prime | p || k } ) |
51 |
|
prmex |
|- Prime e. _V |
52 |
51
|
rabex |
|- { p e. Prime | p || k } e. _V |
53 |
50 41 52
|
fvmpt |
|- ( 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 } ) |
54 |
53
|
adantl |
|- ( ( 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 } ) |
55 |
|
neg1cn |
|- -u 1 e. CC |
56 |
|
prmdvdsfi |
|- ( N e. NN -> { p e. Prime | p || N } e. Fin ) |
57 |
|
elpwi |
|- ( x e. ~P { p e. Prime | p || N } -> x C_ { p e. Prime | p || N } ) |
58 |
|
ssfi |
|- ( ( { p e. Prime | p || N } e. Fin /\ x C_ { p e. Prime | p || N } ) -> x e. Fin ) |
59 |
56 57 58
|
syl2an |
|- ( ( N e. NN /\ x e. ~P { p e. Prime | p || N } ) -> x e. Fin ) |
60 |
|
hashcl |
|- ( x e. Fin -> ( # ` x ) e. NN0 ) |
61 |
59 60
|
syl |
|- ( ( N e. NN /\ x e. ~P { p e. Prime | p || N } ) -> ( # ` x ) e. NN0 ) |
62 |
|
expcl |
|- ( ( -u 1 e. CC /\ ( # ` x ) e. NN0 ) -> ( -u 1 ^ ( # ` x ) ) e. CC ) |
63 |
55 61 62
|
sylancr |
|- ( ( N e. NN /\ x e. ~P { p e. Prime | p || N } ) -> ( -u 1 ^ ( # ` x ) ) e. CC ) |
64 |
38 39 48 54 63
|
fsumf1o |
|- ( 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 } ) ) ) |
65 |
|
fzfid |
|- ( N e. NN -> ( 0 ... ( # ` { p e. Prime | p || N } ) ) e. Fin ) |
66 |
56
|
adantr |
|- ( ( N e. NN /\ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) -> { p e. Prime | p || N } e. Fin ) |
67 |
|
pwfi |
|- ( { p e. Prime | p || N } e. Fin <-> ~P { p e. Prime | p || N } e. Fin ) |
68 |
66 67
|
sylib |
|- ( ( N e. NN /\ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) -> ~P { p e. Prime | p || N } e. Fin ) |
69 |
|
ssrab2 |
|- { s e. ~P { p e. Prime | p || N } | ( # ` s ) = z } C_ ~P { p e. Prime | p || N } |
70 |
|
ssfi |
|- ( ( ~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 ) |
71 |
68 69 70
|
sylancl |
|- ( ( N e. NN /\ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) -> { s e. ~P { p e. Prime | p || N } | ( # ` s ) = z } e. Fin ) |
72 |
|
simprr |
|- ( ( 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 } ) |
73 |
|
fveqeq2 |
|- ( s = x -> ( ( # ` s ) = z <-> ( # ` x ) = z ) ) |
74 |
73
|
elrab |
|- ( x e. { s e. ~P { p e. Prime | p || N } | ( # ` s ) = z } <-> ( x e. ~P { p e. Prime | p || N } /\ ( # ` x ) = z ) ) |
75 |
74
|
simprbi |
|- ( x e. { s e. ~P { p e. Prime | p || N } | ( # ` s ) = z } -> ( # ` x ) = z ) |
76 |
72 75
|
syl |
|- ( ( 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 ) |
77 |
76
|
ralrimivva |
|- ( 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 ) |
78 |
|
invdisj |
|- ( 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 } ) |
79 |
77 78
|
syl |
|- ( N e. NN -> Disj_ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) { s e. ~P { p e. Prime | p || N } | ( # ` s ) = z } ) |
80 |
56
|
adantr |
|- ( ( 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 ) |
81 |
69 72
|
sselid |
|- ( ( 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 } ) |
82 |
81 57
|
syl |
|- ( ( 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 } ) |
83 |
80 82
|
ssfid |
|- ( ( 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 ) |
84 |
83 60
|
syl |
|- ( ( 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 ) |
85 |
55 84 62
|
sylancr |
|- ( ( 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 ) |
86 |
65 71 79 85
|
fsumiun |
|- ( 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 ) ) ) |
87 |
|
iunrab |
|- 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 } |
88 |
56
|
adantr |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> { p e. Prime | p || N } e. Fin ) |
89 |
|
elpwi |
|- ( s e. ~P { p e. Prime | p || N } -> s C_ { p e. Prime | p || N } ) |
90 |
89
|
adantl |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> s C_ { p e. Prime | p || N } ) |
91 |
|
ssdomg |
|- ( { p e. Prime | p || N } e. Fin -> ( s C_ { p e. Prime | p || N } -> s ~<_ { p e. Prime | p || N } ) ) |
92 |
88 90 91
|
sylc |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> s ~<_ { p e. Prime | p || N } ) |
93 |
|
ssfi |
|- ( ( { p e. Prime | p || N } e. Fin /\ s C_ { p e. Prime | p || N } ) -> s e. Fin ) |
94 |
56 89 93
|
syl2an |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> s e. Fin ) |
95 |
|
hashdom |
|- ( ( s e. Fin /\ { p e. Prime | p || N } e. Fin ) -> ( ( # ` s ) <_ ( # ` { p e. Prime | p || N } ) <-> s ~<_ { p e. Prime | p || N } ) ) |
96 |
94 88 95
|
syl2anc |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( ( # ` s ) <_ ( # ` { p e. Prime | p || N } ) <-> s ~<_ { p e. Prime | p || N } ) ) |
97 |
92 96
|
mpbird |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` s ) <_ ( # ` { p e. Prime | p || N } ) ) |
98 |
|
hashcl |
|- ( s e. Fin -> ( # ` s ) e. NN0 ) |
99 |
94 98
|
syl |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` s ) e. NN0 ) |
100 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
101 |
99 100
|
eleqtrdi |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` s ) e. ( ZZ>= ` 0 ) ) |
102 |
|
hashcl |
|- ( { p e. Prime | p || N } e. Fin -> ( # ` { p e. Prime | p || N } ) e. NN0 ) |
103 |
56 102
|
syl |
|- ( N e. NN -> ( # ` { p e. Prime | p || N } ) e. NN0 ) |
104 |
103
|
adantr |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` { p e. Prime | p || N } ) e. NN0 ) |
105 |
104
|
nn0zd |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` { p e. Prime | p || N } ) e. ZZ ) |
106 |
|
elfz5 |
|- ( ( ( # ` 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 } ) ) ) |
107 |
101 105 106
|
syl2anc |
|- ( ( 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 } ) ) ) |
108 |
97 107
|
mpbird |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` s ) e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) |
109 |
|
eqidd |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> ( # ` s ) = ( # ` s ) ) |
110 |
|
eqeq2 |
|- ( z = ( # ` s ) -> ( ( # ` s ) = z <-> ( # ` s ) = ( # ` s ) ) ) |
111 |
110
|
rspcev |
|- ( ( ( # ` s ) e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) /\ ( # ` s ) = ( # ` s ) ) -> E. z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ( # ` s ) = z ) |
112 |
108 109 111
|
syl2anc |
|- ( ( N e. NN /\ s e. ~P { p e. Prime | p || N } ) -> E. z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ( # ` s ) = z ) |
113 |
112
|
ralrimiva |
|- ( N e. NN -> A. s e. ~P { p e. Prime | p || N } E. z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ( # ` s ) = z ) |
114 |
|
rabid2 |
|- ( ~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 ) |
115 |
113 114
|
sylibr |
|- ( 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 } ) |
116 |
87 115
|
eqtr4id |
|- ( 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 } ) |
117 |
116
|
sumeq1d |
|- ( 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 ) ) ) |
118 |
|
elfznn0 |
|- ( z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) -> z e. NN0 ) |
119 |
118
|
adantl |
|- ( ( N e. NN /\ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) -> z e. NN0 ) |
120 |
|
expcl |
|- ( ( -u 1 e. CC /\ z e. NN0 ) -> ( -u 1 ^ z ) e. CC ) |
121 |
55 119 120
|
sylancr |
|- ( ( N e. NN /\ z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) ) -> ( -u 1 ^ z ) e. CC ) |
122 |
|
fsumconst |
|- ( ( { 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 ) ) ) |
123 |
71 121 122
|
syl2anc |
|- ( ( 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 ) ) ) |
124 |
75
|
adantl |
|- ( ( ( 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 ) |
125 |
124
|
oveq2d |
|- ( ( ( 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 ) ) |
126 |
125
|
sumeq2dv |
|- ( ( 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 ) ) |
127 |
|
elfzelz |
|- ( z e. ( 0 ... ( # ` { p e. Prime | p || N } ) ) -> z e. ZZ ) |
128 |
|
hashbc |
|- ( ( { 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 } ) ) |
129 |
56 127 128
|
syl2an |
|- ( ( 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 } ) ) |
130 |
129
|
oveq1d |
|- ( ( 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 ) ) ) |
131 |
123 126 130
|
3eqtr4d |
|- ( ( 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 ) ) ) |
132 |
131
|
sumeq2dv |
|- ( 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 ) ) ) |
133 |
|
1pneg1e0 |
|- ( 1 + -u 1 ) = 0 |
134 |
133
|
oveq1i |
|- ( ( 1 + -u 1 ) ^ ( # ` { p e. Prime | p || N } ) ) = ( 0 ^ ( # ` { p e. Prime | p || N } ) ) |
135 |
|
binom1p |
|- ( ( -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 ) ) ) |
136 |
55 103 135
|
sylancr |
|- ( 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 ) ) ) |
137 |
134 136
|
eqtr3id |
|- ( 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 ) ) ) |
138 |
|
eqeq2 |
|- ( 1 = if ( N = 1 , 1 , 0 ) -> ( ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = 1 <-> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = if ( N = 1 , 1 , 0 ) ) ) |
139 |
|
eqeq2 |
|- ( 0 = if ( N = 1 , 1 , 0 ) -> ( ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = 0 <-> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = if ( N = 1 , 1 , 0 ) ) ) |
140 |
|
nprmdvds1 |
|- ( p e. Prime -> -. p || 1 ) |
141 |
|
simpr |
|- ( ( N e. NN /\ N = 1 ) -> N = 1 ) |
142 |
141
|
breq2d |
|- ( ( N e. NN /\ N = 1 ) -> ( p || N <-> p || 1 ) ) |
143 |
142
|
notbid |
|- ( ( N e. NN /\ N = 1 ) -> ( -. p || N <-> -. p || 1 ) ) |
144 |
140 143
|
syl5ibr |
|- ( ( N e. NN /\ N = 1 ) -> ( p e. Prime -> -. p || N ) ) |
145 |
144
|
ralrimiv |
|- ( ( N e. NN /\ N = 1 ) -> A. p e. Prime -. p || N ) |
146 |
|
rabeq0 |
|- ( { p e. Prime | p || N } = (/) <-> A. p e. Prime -. p || N ) |
147 |
145 146
|
sylibr |
|- ( ( N e. NN /\ N = 1 ) -> { p e. Prime | p || N } = (/) ) |
148 |
147
|
fveq2d |
|- ( ( N e. NN /\ N = 1 ) -> ( # ` { p e. Prime | p || N } ) = ( # ` (/) ) ) |
149 |
|
hash0 |
|- ( # ` (/) ) = 0 |
150 |
148 149
|
eqtrdi |
|- ( ( N e. NN /\ N = 1 ) -> ( # ` { p e. Prime | p || N } ) = 0 ) |
151 |
150
|
oveq2d |
|- ( ( N e. NN /\ N = 1 ) -> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = ( 0 ^ 0 ) ) |
152 |
|
0exp0e1 |
|- ( 0 ^ 0 ) = 1 |
153 |
151 152
|
eqtrdi |
|- ( ( N e. NN /\ N = 1 ) -> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = 1 ) |
154 |
|
df-ne |
|- ( N =/= 1 <-> -. N = 1 ) |
155 |
|
eluz2b3 |
|- ( N e. ( ZZ>= ` 2 ) <-> ( N e. NN /\ N =/= 1 ) ) |
156 |
155
|
biimpri |
|- ( ( N e. NN /\ N =/= 1 ) -> N e. ( ZZ>= ` 2 ) ) |
157 |
154 156
|
sylan2br |
|- ( ( N e. NN /\ -. N = 1 ) -> N e. ( ZZ>= ` 2 ) ) |
158 |
|
exprmfct |
|- ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N ) |
159 |
157 158
|
syl |
|- ( ( N e. NN /\ -. N = 1 ) -> E. p e. Prime p || N ) |
160 |
|
rabn0 |
|- ( { p e. Prime | p || N } =/= (/) <-> E. p e. Prime p || N ) |
161 |
159 160
|
sylibr |
|- ( ( N e. NN /\ -. N = 1 ) -> { p e. Prime | p || N } =/= (/) ) |
162 |
56
|
adantr |
|- ( ( N e. NN /\ -. N = 1 ) -> { p e. Prime | p || N } e. Fin ) |
163 |
|
hashnncl |
|- ( { p e. Prime | p || N } e. Fin -> ( ( # ` { p e. Prime | p || N } ) e. NN <-> { p e. Prime | p || N } =/= (/) ) ) |
164 |
162 163
|
syl |
|- ( ( N e. NN /\ -. N = 1 ) -> ( ( # ` { p e. Prime | p || N } ) e. NN <-> { p e. Prime | p || N } =/= (/) ) ) |
165 |
161 164
|
mpbird |
|- ( ( N e. NN /\ -. N = 1 ) -> ( # ` { p e. Prime | p || N } ) e. NN ) |
166 |
165
|
0expd |
|- ( ( N e. NN /\ -. N = 1 ) -> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = 0 ) |
167 |
138 139 153 166
|
ifbothda |
|- ( N e. NN -> ( 0 ^ ( # ` { p e. Prime | p || N } ) ) = if ( N = 1 , 1 , 0 ) ) |
168 |
132 137 167
|
3eqtr2d |
|- ( 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 ) ) |
169 |
86 117 168
|
3eqtr3d |
|- ( N e. NN -> sum_ x e. ~P { p e. Prime | p || N } ( -u 1 ^ ( # ` x ) ) = if ( N = 1 , 1 , 0 ) ) |
170 |
64 169
|
eqtr3d |
|- ( 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 ) ) |
171 |
10 36 170
|
3eqtr3d |
|- ( N e. NN -> sum_ k e. { n e. NN | n || N } ( mmu ` k ) = if ( N = 1 , 1 , 0 ) ) |