Step |
Hyp |
Ref |
Expression |
1 |
|
muinv.1 |
|- ( ph -> F : NN --> CC ) |
2 |
|
muinv.2 |
|- ( ph -> G = ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) ) |
3 |
1
|
feqmptd |
|- ( ph -> F = ( m e. NN |-> ( F ` m ) ) ) |
4 |
2
|
ad2antrr |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> G = ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) ) |
5 |
4
|
fveq1d |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( G ` ( m / j ) ) = ( ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) ` ( m / j ) ) ) |
6 |
|
breq1 |
|- ( x = j -> ( x || m <-> j || m ) ) |
7 |
6
|
elrab |
|- ( j e. { x e. NN | x || m } <-> ( j e. NN /\ j || m ) ) |
8 |
7
|
simprbi |
|- ( j e. { x e. NN | x || m } -> j || m ) |
9 |
8
|
adantl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> j || m ) |
10 |
|
elrabi |
|- ( j e. { x e. NN | x || m } -> j e. NN ) |
11 |
10
|
adantl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> j e. NN ) |
12 |
11
|
nnzd |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> j e. ZZ ) |
13 |
11
|
nnne0d |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> j =/= 0 ) |
14 |
|
nnz |
|- ( m e. NN -> m e. ZZ ) |
15 |
14
|
ad2antlr |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> m e. ZZ ) |
16 |
|
dvdsval2 |
|- ( ( j e. ZZ /\ j =/= 0 /\ m e. ZZ ) -> ( j || m <-> ( m / j ) e. ZZ ) ) |
17 |
12 13 15 16
|
syl3anc |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( j || m <-> ( m / j ) e. ZZ ) ) |
18 |
9 17
|
mpbid |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( m / j ) e. ZZ ) |
19 |
|
nnre |
|- ( m e. NN -> m e. RR ) |
20 |
|
nngt0 |
|- ( m e. NN -> 0 < m ) |
21 |
19 20
|
jca |
|- ( m e. NN -> ( m e. RR /\ 0 < m ) ) |
22 |
21
|
ad2antlr |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( m e. RR /\ 0 < m ) ) |
23 |
|
nnre |
|- ( j e. NN -> j e. RR ) |
24 |
|
nngt0 |
|- ( j e. NN -> 0 < j ) |
25 |
23 24
|
jca |
|- ( j e. NN -> ( j e. RR /\ 0 < j ) ) |
26 |
11 25
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( j e. RR /\ 0 < j ) ) |
27 |
|
divgt0 |
|- ( ( ( m e. RR /\ 0 < m ) /\ ( j e. RR /\ 0 < j ) ) -> 0 < ( m / j ) ) |
28 |
22 26 27
|
syl2anc |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> 0 < ( m / j ) ) |
29 |
|
elnnz |
|- ( ( m / j ) e. NN <-> ( ( m / j ) e. ZZ /\ 0 < ( m / j ) ) ) |
30 |
18 28 29
|
sylanbrc |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( m / j ) e. NN ) |
31 |
|
breq2 |
|- ( n = ( m / j ) -> ( x || n <-> x || ( m / j ) ) ) |
32 |
31
|
rabbidv |
|- ( n = ( m / j ) -> { x e. NN | x || n } = { x e. NN | x || ( m / j ) } ) |
33 |
32
|
sumeq1d |
|- ( n = ( m / j ) -> sum_ k e. { x e. NN | x || n } ( F ` k ) = sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) |
34 |
|
eqid |
|- ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) = ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) |
35 |
|
sumex |
|- sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) e. _V |
36 |
33 34 35
|
fvmpt |
|- ( ( m / j ) e. NN -> ( ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) ` ( m / j ) ) = sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) |
37 |
30 36
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( ( n e. NN |-> sum_ k e. { x e. NN | x || n } ( F ` k ) ) ` ( m / j ) ) = sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) |
38 |
5 37
|
eqtrd |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( G ` ( m / j ) ) = sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) |
39 |
38
|
oveq2d |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) = ( ( mmu ` j ) x. sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) ) |
40 |
|
fzfid |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( 1 ... ( m / j ) ) e. Fin ) |
41 |
|
dvdsssfz1 |
|- ( ( m / j ) e. NN -> { x e. NN | x || ( m / j ) } C_ ( 1 ... ( m / j ) ) ) |
42 |
30 41
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> { x e. NN | x || ( m / j ) } C_ ( 1 ... ( m / j ) ) ) |
43 |
40 42
|
ssfid |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> { x e. NN | x || ( m / j ) } e. Fin ) |
44 |
|
mucl |
|- ( j e. NN -> ( mmu ` j ) e. ZZ ) |
45 |
11 44
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( mmu ` j ) e. ZZ ) |
46 |
45
|
zcnd |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( mmu ` j ) e. CC ) |
47 |
1
|
ad2antrr |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> F : NN --> CC ) |
48 |
|
elrabi |
|- ( k e. { x e. NN | x || ( m / j ) } -> k e. NN ) |
49 |
|
ffvelrn |
|- ( ( F : NN --> CC /\ k e. NN ) -> ( F ` k ) e. CC ) |
50 |
47 48 49
|
syl2an |
|- ( ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) /\ k e. { x e. NN | x || ( m / j ) } ) -> ( F ` k ) e. CC ) |
51 |
43 46 50
|
fsummulc2 |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( ( mmu ` j ) x. sum_ k e. { x e. NN | x || ( m / j ) } ( F ` k ) ) = sum_ k e. { x e. NN | x || ( m / j ) } ( ( mmu ` j ) x. ( F ` k ) ) ) |
52 |
39 51
|
eqtrd |
|- ( ( ( ph /\ m e. NN ) /\ j e. { x e. NN | x || m } ) -> ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) = sum_ k e. { x e. NN | x || ( m / j ) } ( ( mmu ` j ) x. ( F ` k ) ) ) |
53 |
52
|
sumeq2dv |
|- ( ( ph /\ m e. NN ) -> sum_ j e. { x e. NN | x || m } ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) = sum_ j e. { x e. NN | x || m } sum_ k e. { x e. NN | x || ( m / j ) } ( ( mmu ` j ) x. ( F ` k ) ) ) |
54 |
|
simpr |
|- ( ( ph /\ m e. NN ) -> m e. NN ) |
55 |
46
|
adantrr |
|- ( ( ( ph /\ m e. NN ) /\ ( j e. { x e. NN | x || m } /\ k e. { x e. NN | x || ( m / j ) } ) ) -> ( mmu ` j ) e. CC ) |
56 |
50
|
anasss |
|- ( ( ( ph /\ m e. NN ) /\ ( j e. { x e. NN | x || m } /\ k e. { x e. NN | x || ( m / j ) } ) ) -> ( F ` k ) e. CC ) |
57 |
55 56
|
mulcld |
|- ( ( ( ph /\ m e. NN ) /\ ( j e. { x e. NN | x || m } /\ k e. { x e. NN | x || ( m / j ) } ) ) -> ( ( mmu ` j ) x. ( F ` k ) ) e. CC ) |
58 |
54 57
|
fsumdvdsdiag |
|- ( ( ph /\ m e. NN ) -> sum_ j e. { x e. NN | x || m } sum_ k e. { x e. NN | x || ( m / j ) } ( ( mmu ` j ) x. ( F ` k ) ) = sum_ k e. { x e. NN | x || m } sum_ j e. { x e. NN | x || ( m / k ) } ( ( mmu ` j ) x. ( F ` k ) ) ) |
59 |
|
ssrab2 |
|- { x e. NN | x || m } C_ NN |
60 |
|
dvdsdivcl |
|- ( ( m e. NN /\ k e. { x e. NN | x || m } ) -> ( m / k ) e. { x e. NN | x || m } ) |
61 |
60
|
adantll |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( m / k ) e. { x e. NN | x || m } ) |
62 |
59 61
|
sselid |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( m / k ) e. NN ) |
63 |
|
musum |
|- ( ( m / k ) e. NN -> sum_ j e. { x e. NN | x || ( m / k ) } ( mmu ` j ) = if ( ( m / k ) = 1 , 1 , 0 ) ) |
64 |
62 63
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> sum_ j e. { x e. NN | x || ( m / k ) } ( mmu ` j ) = if ( ( m / k ) = 1 , 1 , 0 ) ) |
65 |
64
|
oveq1d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( sum_ j e. { x e. NN | x || ( m / k ) } ( mmu ` j ) x. ( F ` k ) ) = ( if ( ( m / k ) = 1 , 1 , 0 ) x. ( F ` k ) ) ) |
66 |
|
fzfid |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( 1 ... ( m / k ) ) e. Fin ) |
67 |
|
dvdsssfz1 |
|- ( ( m / k ) e. NN -> { x e. NN | x || ( m / k ) } C_ ( 1 ... ( m / k ) ) ) |
68 |
62 67
|
syl |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> { x e. NN | x || ( m / k ) } C_ ( 1 ... ( m / k ) ) ) |
69 |
66 68
|
ssfid |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> { x e. NN | x || ( m / k ) } e. Fin ) |
70 |
1
|
adantr |
|- ( ( ph /\ m e. NN ) -> F : NN --> CC ) |
71 |
|
elrabi |
|- ( k e. { x e. NN | x || m } -> k e. NN ) |
72 |
70 71 49
|
syl2an |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( F ` k ) e. CC ) |
73 |
|
ssrab2 |
|- { x e. NN | x || ( m / k ) } C_ NN |
74 |
|
simpr |
|- ( ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) /\ j e. { x e. NN | x || ( m / k ) } ) -> j e. { x e. NN | x || ( m / k ) } ) |
75 |
73 74
|
sselid |
|- ( ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) /\ j e. { x e. NN | x || ( m / k ) } ) -> j e. NN ) |
76 |
75 44
|
syl |
|- ( ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) /\ j e. { x e. NN | x || ( m / k ) } ) -> ( mmu ` j ) e. ZZ ) |
77 |
76
|
zcnd |
|- ( ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) /\ j e. { x e. NN | x || ( m / k ) } ) -> ( mmu ` j ) e. CC ) |
78 |
69 72 77
|
fsummulc1 |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( sum_ j e. { x e. NN | x || ( m / k ) } ( mmu ` j ) x. ( F ` k ) ) = sum_ j e. { x e. NN | x || ( m / k ) } ( ( mmu ` j ) x. ( F ` k ) ) ) |
79 |
|
ovif |
|- ( if ( ( m / k ) = 1 , 1 , 0 ) x. ( F ` k ) ) = if ( ( m / k ) = 1 , ( 1 x. ( F ` k ) ) , ( 0 x. ( F ` k ) ) ) |
80 |
|
nncn |
|- ( m e. NN -> m e. CC ) |
81 |
80
|
ad2antlr |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> m e. CC ) |
82 |
71
|
adantl |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> k e. NN ) |
83 |
82
|
nncnd |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> k e. CC ) |
84 |
|
1cnd |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> 1 e. CC ) |
85 |
82
|
nnne0d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> k =/= 0 ) |
86 |
81 83 84 85
|
divmuld |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( ( m / k ) = 1 <-> ( k x. 1 ) = m ) ) |
87 |
83
|
mulid1d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( k x. 1 ) = k ) |
88 |
87
|
eqeq1d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( ( k x. 1 ) = m <-> k = m ) ) |
89 |
86 88
|
bitrd |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( ( m / k ) = 1 <-> k = m ) ) |
90 |
72
|
mulid2d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( 1 x. ( F ` k ) ) = ( F ` k ) ) |
91 |
72
|
mul02d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( 0 x. ( F ` k ) ) = 0 ) |
92 |
89 90 91
|
ifbieq12d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> if ( ( m / k ) = 1 , ( 1 x. ( F ` k ) ) , ( 0 x. ( F ` k ) ) ) = if ( k = m , ( F ` k ) , 0 ) ) |
93 |
79 92
|
syl5eq |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> ( if ( ( m / k ) = 1 , 1 , 0 ) x. ( F ` k ) ) = if ( k = m , ( F ` k ) , 0 ) ) |
94 |
65 78 93
|
3eqtr3d |
|- ( ( ( ph /\ m e. NN ) /\ k e. { x e. NN | x || m } ) -> sum_ j e. { x e. NN | x || ( m / k ) } ( ( mmu ` j ) x. ( F ` k ) ) = if ( k = m , ( F ` k ) , 0 ) ) |
95 |
94
|
sumeq2dv |
|- ( ( ph /\ m e. NN ) -> sum_ k e. { x e. NN | x || m } sum_ j e. { x e. NN | x || ( m / k ) } ( ( mmu ` j ) x. ( F ` k ) ) = sum_ k e. { x e. NN | x || m } if ( k = m , ( F ` k ) , 0 ) ) |
96 |
|
breq1 |
|- ( x = m -> ( x || m <-> m || m ) ) |
97 |
54
|
nnzd |
|- ( ( ph /\ m e. NN ) -> m e. ZZ ) |
98 |
|
iddvds |
|- ( m e. ZZ -> m || m ) |
99 |
97 98
|
syl |
|- ( ( ph /\ m e. NN ) -> m || m ) |
100 |
96 54 99
|
elrabd |
|- ( ( ph /\ m e. NN ) -> m e. { x e. NN | x || m } ) |
101 |
100
|
snssd |
|- ( ( ph /\ m e. NN ) -> { m } C_ { x e. NN | x || m } ) |
102 |
101
|
sselda |
|- ( ( ( ph /\ m e. NN ) /\ k e. { m } ) -> k e. { x e. NN | x || m } ) |
103 |
102 72
|
syldan |
|- ( ( ( ph /\ m e. NN ) /\ k e. { m } ) -> ( F ` k ) e. CC ) |
104 |
|
0cn |
|- 0 e. CC |
105 |
|
ifcl |
|- ( ( ( F ` k ) e. CC /\ 0 e. CC ) -> if ( k = m , ( F ` k ) , 0 ) e. CC ) |
106 |
103 104 105
|
sylancl |
|- ( ( ( ph /\ m e. NN ) /\ k e. { m } ) -> if ( k = m , ( F ` k ) , 0 ) e. CC ) |
107 |
|
eldifsni |
|- ( k e. ( { x e. NN | x || m } \ { m } ) -> k =/= m ) |
108 |
107
|
adantl |
|- ( ( ( ph /\ m e. NN ) /\ k e. ( { x e. NN | x || m } \ { m } ) ) -> k =/= m ) |
109 |
108
|
neneqd |
|- ( ( ( ph /\ m e. NN ) /\ k e. ( { x e. NN | x || m } \ { m } ) ) -> -. k = m ) |
110 |
109
|
iffalsed |
|- ( ( ( ph /\ m e. NN ) /\ k e. ( { x e. NN | x || m } \ { m } ) ) -> if ( k = m , ( F ` k ) , 0 ) = 0 ) |
111 |
|
fzfid |
|- ( ( ph /\ m e. NN ) -> ( 1 ... m ) e. Fin ) |
112 |
|
dvdsssfz1 |
|- ( m e. NN -> { x e. NN | x || m } C_ ( 1 ... m ) ) |
113 |
112
|
adantl |
|- ( ( ph /\ m e. NN ) -> { x e. NN | x || m } C_ ( 1 ... m ) ) |
114 |
111 113
|
ssfid |
|- ( ( ph /\ m e. NN ) -> { x e. NN | x || m } e. Fin ) |
115 |
101 106 110 114
|
fsumss |
|- ( ( ph /\ m e. NN ) -> sum_ k e. { m } if ( k = m , ( F ` k ) , 0 ) = sum_ k e. { x e. NN | x || m } if ( k = m , ( F ` k ) , 0 ) ) |
116 |
1
|
ffvelrnda |
|- ( ( ph /\ m e. NN ) -> ( F ` m ) e. CC ) |
117 |
|
iftrue |
|- ( k = m -> if ( k = m , ( F ` k ) , 0 ) = ( F ` k ) ) |
118 |
|
fveq2 |
|- ( k = m -> ( F ` k ) = ( F ` m ) ) |
119 |
117 118
|
eqtrd |
|- ( k = m -> if ( k = m , ( F ` k ) , 0 ) = ( F ` m ) ) |
120 |
119
|
sumsn |
|- ( ( m e. NN /\ ( F ` m ) e. CC ) -> sum_ k e. { m } if ( k = m , ( F ` k ) , 0 ) = ( F ` m ) ) |
121 |
54 116 120
|
syl2anc |
|- ( ( ph /\ m e. NN ) -> sum_ k e. { m } if ( k = m , ( F ` k ) , 0 ) = ( F ` m ) ) |
122 |
95 115 121
|
3eqtr2d |
|- ( ( ph /\ m e. NN ) -> sum_ k e. { x e. NN | x || m } sum_ j e. { x e. NN | x || ( m / k ) } ( ( mmu ` j ) x. ( F ` k ) ) = ( F ` m ) ) |
123 |
53 58 122
|
3eqtrd |
|- ( ( ph /\ m e. NN ) -> sum_ j e. { x e. NN | x || m } ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) = ( F ` m ) ) |
124 |
123
|
mpteq2dva |
|- ( ph -> ( m e. NN |-> sum_ j e. { x e. NN | x || m } ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) ) = ( m e. NN |-> ( F ` m ) ) ) |
125 |
3 124
|
eqtr4d |
|- ( ph -> F = ( m e. NN |-> sum_ j e. { x e. NN | x || m } ( ( mmu ` j ) x. ( G ` ( m / j ) ) ) ) ) |