Step |
Hyp |
Ref |
Expression |
1 |
|
fprodser.1 |
|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) = A ) |
2 |
|
fprodser.2 |
|- ( ph -> N e. ( ZZ>= ` M ) ) |
3 |
|
fprodser.3 |
|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC ) |
4 |
|
prodfc |
|- prod_ j e. ( M ... N ) ( ( k e. ( M ... N ) |-> A ) ` j ) = prod_ k e. ( M ... N ) A |
5 |
|
fveq2 |
|- ( j = ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) -> ( ( k e. ( M ... N ) |-> A ) ` j ) = ( ( k e. ( M ... N ) |-> A ) ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) ) |
6 |
|
eluzelz |
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) |
7 |
2 6
|
syl |
|- ( ph -> N e. ZZ ) |
8 |
7
|
zcnd |
|- ( ph -> N e. CC ) |
9 |
|
eluzel2 |
|- ( N e. ( ZZ>= ` M ) -> M e. ZZ ) |
10 |
2 9
|
syl |
|- ( ph -> M e. ZZ ) |
11 |
10
|
zcnd |
|- ( ph -> M e. CC ) |
12 |
|
1cnd |
|- ( ph -> 1 e. CC ) |
13 |
8 11 12
|
subadd23d |
|- ( ph -> ( ( N - M ) + 1 ) = ( N + ( 1 - M ) ) ) |
14 |
13
|
eqcomd |
|- ( ph -> ( N + ( 1 - M ) ) = ( ( N - M ) + 1 ) ) |
15 |
|
uznn0sub |
|- ( N e. ( ZZ>= ` M ) -> ( N - M ) e. NN0 ) |
16 |
2 15
|
syl |
|- ( ph -> ( N - M ) e. NN0 ) |
17 |
|
nn0p1nn |
|- ( ( N - M ) e. NN0 -> ( ( N - M ) + 1 ) e. NN ) |
18 |
16 17
|
syl |
|- ( ph -> ( ( N - M ) + 1 ) e. NN ) |
19 |
14 18
|
eqeltrd |
|- ( ph -> ( N + ( 1 - M ) ) e. NN ) |
20 |
12 11
|
pncan3d |
|- ( ph -> ( 1 + ( M - 1 ) ) = M ) |
21 |
8 12 11
|
pnpncand |
|- ( ph -> ( ( N + ( 1 - M ) ) + ( M - 1 ) ) = N ) |
22 |
20 21
|
oveq12d |
|- ( ph -> ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) = ( M ... N ) ) |
23 |
22
|
eleq2d |
|- ( ph -> ( p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) <-> p e. ( M ... N ) ) ) |
24 |
23
|
biimpa |
|- ( ( ph /\ p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) -> p e. ( M ... N ) ) |
25 |
|
elfzelz |
|- ( p e. ( M ... N ) -> p e. ZZ ) |
26 |
25
|
zcnd |
|- ( p e. ( M ... N ) -> p e. CC ) |
27 |
26
|
adantl |
|- ( ( ph /\ p e. ( M ... N ) ) -> p e. CC ) |
28 |
|
peano2zm |
|- ( M e. ZZ -> ( M - 1 ) e. ZZ ) |
29 |
10 28
|
syl |
|- ( ph -> ( M - 1 ) e. ZZ ) |
30 |
29
|
zcnd |
|- ( ph -> ( M - 1 ) e. CC ) |
31 |
30
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( M - 1 ) e. CC ) |
32 |
27 31
|
npcand |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( ( p - ( M - 1 ) ) + ( M - 1 ) ) = p ) |
33 |
|
simpr |
|- ( ( ph /\ p e. ( M ... N ) ) -> p e. ( M ... N ) ) |
34 |
32 33
|
eqeltrd |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( ( p - ( M - 1 ) ) + ( M - 1 ) ) e. ( M ... N ) ) |
35 |
|
ovex |
|- ( p - ( M - 1 ) ) e. _V |
36 |
|
oveq1 |
|- ( n = ( p - ( M - 1 ) ) -> ( n + ( M - 1 ) ) = ( ( p - ( M - 1 ) ) + ( M - 1 ) ) ) |
37 |
36
|
eleq1d |
|- ( n = ( p - ( M - 1 ) ) -> ( ( n + ( M - 1 ) ) e. ( M ... N ) <-> ( ( p - ( M - 1 ) ) + ( M - 1 ) ) e. ( M ... N ) ) ) |
38 |
35 37
|
sbcie |
|- ( [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) <-> ( ( p - ( M - 1 ) ) + ( M - 1 ) ) e. ( M ... N ) ) |
39 |
34 38
|
sylibr |
|- ( ( ph /\ p e. ( M ... N ) ) -> [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) ) |
40 |
24 39
|
syldan |
|- ( ( ph /\ p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) -> [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) ) |
41 |
40
|
ralrimiva |
|- ( ph -> A. p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) ) |
42 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
43 |
19
|
nnzd |
|- ( ph -> ( N + ( 1 - M ) ) e. ZZ ) |
44 |
|
fzshftral |
|- ( ( 1 e. ZZ /\ ( N + ( 1 - M ) ) e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( A. n e. ( 1 ... ( N + ( 1 - M ) ) ) ( n + ( M - 1 ) ) e. ( M ... N ) <-> A. p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) ) ) |
45 |
42 43 29 44
|
syl3anc |
|- ( ph -> ( A. n e. ( 1 ... ( N + ( 1 - M ) ) ) ( n + ( M - 1 ) ) e. ( M ... N ) <-> A. p e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) [. ( p - ( M - 1 ) ) / n ]. ( n + ( M - 1 ) ) e. ( M ... N ) ) ) |
46 |
41 45
|
mpbird |
|- ( ph -> A. n e. ( 1 ... ( N + ( 1 - M ) ) ) ( n + ( M - 1 ) ) e. ( M ... N ) ) |
47 |
10
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> M e. ZZ ) |
48 |
7
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> N e. ZZ ) |
49 |
25
|
adantl |
|- ( ( ph /\ p e. ( M ... N ) ) -> p e. ZZ ) |
50 |
29
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( M - 1 ) e. ZZ ) |
51 |
|
fzsubel |
|- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( p e. ZZ /\ ( M - 1 ) e. ZZ ) ) -> ( p e. ( M ... N ) <-> ( p - ( M - 1 ) ) e. ( ( M - ( M - 1 ) ) ... ( N - ( M - 1 ) ) ) ) ) |
52 |
47 48 49 50 51
|
syl22anc |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( p e. ( M ... N ) <-> ( p - ( M - 1 ) ) e. ( ( M - ( M - 1 ) ) ... ( N - ( M - 1 ) ) ) ) ) |
53 |
33 52
|
mpbid |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( p - ( M - 1 ) ) e. ( ( M - ( M - 1 ) ) ... ( N - ( M - 1 ) ) ) ) |
54 |
11 12
|
nncand |
|- ( ph -> ( M - ( M - 1 ) ) = 1 ) |
55 |
8 11 12
|
subsub2d |
|- ( ph -> ( N - ( M - 1 ) ) = ( N + ( 1 - M ) ) ) |
56 |
54 55
|
oveq12d |
|- ( ph -> ( ( M - ( M - 1 ) ) ... ( N - ( M - 1 ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) ) |
57 |
56
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( ( M - ( M - 1 ) ) ... ( N - ( M - 1 ) ) ) = ( 1 ... ( N + ( 1 - M ) ) ) ) |
58 |
53 57
|
eleqtrd |
|- ( ( ph /\ p e. ( M ... N ) ) -> ( p - ( M - 1 ) ) e. ( 1 ... ( N + ( 1 - M ) ) ) ) |
59 |
32
|
eqcomd |
|- ( ( ph /\ p e. ( M ... N ) ) -> p = ( ( p - ( M - 1 ) ) + ( M - 1 ) ) ) |
60 |
36
|
rspceeqv |
|- ( ( ( p - ( M - 1 ) ) e. ( 1 ... ( N + ( 1 - M ) ) ) /\ p = ( ( p - ( M - 1 ) ) + ( M - 1 ) ) ) -> E. n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) ) |
61 |
58 59 60
|
syl2anc |
|- ( ( ph /\ p e. ( M ... N ) ) -> E. n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) ) |
62 |
|
elfzelz |
|- ( n e. ( 1 ... ( N + ( 1 - M ) ) ) -> n e. ZZ ) |
63 |
62
|
zcnd |
|- ( n e. ( 1 ... ( N + ( 1 - M ) ) ) -> n e. CC ) |
64 |
|
elfzelz |
|- ( m e. ( 1 ... ( N + ( 1 - M ) ) ) -> m e. ZZ ) |
65 |
64
|
zcnd |
|- ( m e. ( 1 ... ( N + ( 1 - M ) ) ) -> m e. CC ) |
66 |
63 65
|
anim12i |
|- ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( n e. CC /\ m e. CC ) ) |
67 |
|
eqtr2 |
|- ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> ( n + ( M - 1 ) ) = ( m + ( M - 1 ) ) ) |
68 |
|
simprl |
|- ( ( ph /\ ( n e. CC /\ m e. CC ) ) -> n e. CC ) |
69 |
|
simprr |
|- ( ( ph /\ ( n e. CC /\ m e. CC ) ) -> m e. CC ) |
70 |
30
|
adantr |
|- ( ( ph /\ ( n e. CC /\ m e. CC ) ) -> ( M - 1 ) e. CC ) |
71 |
68 69 70
|
addcan2d |
|- ( ( ph /\ ( n e. CC /\ m e. CC ) ) -> ( ( n + ( M - 1 ) ) = ( m + ( M - 1 ) ) <-> n = m ) ) |
72 |
67 71
|
syl5ib |
|- ( ( ph /\ ( n e. CC /\ m e. CC ) ) -> ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> n = m ) ) |
73 |
66 72
|
sylan2 |
|- ( ( ph /\ ( n e. ( 1 ... ( N + ( 1 - M ) ) ) /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) ) -> ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> n = m ) ) |
74 |
73
|
ralrimivva |
|- ( ph -> A. n e. ( 1 ... ( N + ( 1 - M ) ) ) A. m e. ( 1 ... ( N + ( 1 - M ) ) ) ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> n = m ) ) |
75 |
74
|
adantr |
|- ( ( ph /\ p e. ( M ... N ) ) -> A. n e. ( 1 ... ( N + ( 1 - M ) ) ) A. m e. ( 1 ... ( N + ( 1 - M ) ) ) ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> n = m ) ) |
76 |
|
oveq1 |
|- ( n = m -> ( n + ( M - 1 ) ) = ( m + ( M - 1 ) ) ) |
77 |
76
|
eqeq2d |
|- ( n = m -> ( p = ( n + ( M - 1 ) ) <-> p = ( m + ( M - 1 ) ) ) ) |
78 |
77
|
reu4 |
|- ( E! n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) <-> ( E. n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) /\ A. n e. ( 1 ... ( N + ( 1 - M ) ) ) A. m e. ( 1 ... ( N + ( 1 - M ) ) ) ( ( p = ( n + ( M - 1 ) ) /\ p = ( m + ( M - 1 ) ) ) -> n = m ) ) ) |
79 |
61 75 78
|
sylanbrc |
|- ( ( ph /\ p e. ( M ... N ) ) -> E! n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) ) |
80 |
79
|
ralrimiva |
|- ( ph -> A. p e. ( M ... N ) E! n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) ) |
81 |
|
eqid |
|- ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) = ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) |
82 |
81
|
f1ompt |
|- ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) <-> ( A. n e. ( 1 ... ( N + ( 1 - M ) ) ) ( n + ( M - 1 ) ) e. ( M ... N ) /\ A. p e. ( M ... N ) E! n e. ( 1 ... ( N + ( 1 - M ) ) ) p = ( n + ( M - 1 ) ) ) ) |
83 |
46 80 82
|
sylanbrc |
|- ( ph -> ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) ) |
84 |
3
|
fmpttd |
|- ( ph -> ( k e. ( M ... N ) |-> A ) : ( M ... N ) --> CC ) |
85 |
84
|
ffvelrnda |
|- ( ( ph /\ j e. ( M ... N ) ) -> ( ( k e. ( M ... N ) |-> A ) ` j ) e. CC ) |
86 |
|
simpr |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> m e. ( 1 ... ( N + ( 1 - M ) ) ) ) |
87 |
|
1zzd |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> 1 e. ZZ ) |
88 |
43
|
adantr |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( N + ( 1 - M ) ) e. ZZ ) |
89 |
64
|
adantl |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> m e. ZZ ) |
90 |
29
|
adantr |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( M - 1 ) e. ZZ ) |
91 |
|
fzaddel |
|- ( ( ( 1 e. ZZ /\ ( N + ( 1 - M ) ) e. ZZ ) /\ ( m e. ZZ /\ ( M - 1 ) e. ZZ ) ) -> ( m e. ( 1 ... ( N + ( 1 - M ) ) ) <-> ( m + ( M - 1 ) ) e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) ) |
92 |
87 88 89 90 91
|
syl22anc |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( m e. ( 1 ... ( N + ( 1 - M ) ) ) <-> ( m + ( M - 1 ) ) e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) ) |
93 |
86 92
|
mpbid |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( m + ( M - 1 ) ) e. ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) |
94 |
22
|
adantr |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( 1 + ( M - 1 ) ) ... ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) = ( M ... N ) ) |
95 |
93 94
|
eleqtrd |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( m + ( M - 1 ) ) e. ( M ... N ) ) |
96 |
1
|
ralrimiva |
|- ( ph -> A. k e. ( M ... N ) ( F ` k ) = A ) |
97 |
|
nfcsb1v |
|- F/_ k [_ ( m + ( M - 1 ) ) / k ]_ A |
98 |
97
|
nfeq2 |
|- F/ k ( F ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A |
99 |
|
fveq2 |
|- ( k = ( m + ( M - 1 ) ) -> ( F ` k ) = ( F ` ( m + ( M - 1 ) ) ) ) |
100 |
|
csbeq1a |
|- ( k = ( m + ( M - 1 ) ) -> A = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
101 |
99 100
|
eqeq12d |
|- ( k = ( m + ( M - 1 ) ) -> ( ( F ` k ) = A <-> ( F ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) ) |
102 |
98 101
|
rspc |
|- ( ( m + ( M - 1 ) ) e. ( M ... N ) -> ( A. k e. ( M ... N ) ( F ` k ) = A -> ( F ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) ) |
103 |
96 102
|
mpan9 |
|- ( ( ph /\ ( m + ( M - 1 ) ) e. ( M ... N ) ) -> ( F ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
104 |
95 103
|
syldan |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( F ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
105 |
|
f1of |
|- ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) -1-1-onto-> ( M ... N ) -> ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) --> ( M ... N ) ) |
106 |
83 105
|
syl |
|- ( ph -> ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) --> ( M ... N ) ) |
107 |
|
fvco3 |
|- ( ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) : ( 1 ... ( N + ( 1 - M ) ) ) --> ( M ... N ) /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ` m ) = ( F ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) ) |
108 |
106 107
|
sylan |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ` m ) = ( F ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) ) |
109 |
|
ovex |
|- ( m + ( M - 1 ) ) e. _V |
110 |
76 81 109
|
fvmpt |
|- ( m e. ( 1 ... ( N + ( 1 - M ) ) ) -> ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) = ( m + ( M - 1 ) ) ) |
111 |
110
|
adantl |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) = ( m + ( M - 1 ) ) ) |
112 |
111
|
fveq2d |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( F ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) = ( F ` ( m + ( M - 1 ) ) ) ) |
113 |
108 112
|
eqtrd |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ` m ) = ( F ` ( m + ( M - 1 ) ) ) ) |
114 |
111
|
fveq2d |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( k e. ( M ... N ) |-> A ) ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) = ( ( k e. ( M ... N ) |-> A ) ` ( m + ( M - 1 ) ) ) ) |
115 |
3
|
ralrimiva |
|- ( ph -> A. k e. ( M ... N ) A e. CC ) |
116 |
97
|
nfel1 |
|- F/ k [_ ( m + ( M - 1 ) ) / k ]_ A e. CC |
117 |
100
|
eleq1d |
|- ( k = ( m + ( M - 1 ) ) -> ( A e. CC <-> [_ ( m + ( M - 1 ) ) / k ]_ A e. CC ) ) |
118 |
116 117
|
rspc |
|- ( ( m + ( M - 1 ) ) e. ( M ... N ) -> ( A. k e. ( M ... N ) A e. CC -> [_ ( m + ( M - 1 ) ) / k ]_ A e. CC ) ) |
119 |
115 118
|
mpan9 |
|- ( ( ph /\ ( m + ( M - 1 ) ) e. ( M ... N ) ) -> [_ ( m + ( M - 1 ) ) / k ]_ A e. CC ) |
120 |
95 119
|
syldan |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> [_ ( m + ( M - 1 ) ) / k ]_ A e. CC ) |
121 |
|
eqid |
|- ( k e. ( M ... N ) |-> A ) = ( k e. ( M ... N ) |-> A ) |
122 |
121
|
fvmpts |
|- ( ( ( m + ( M - 1 ) ) e. ( M ... N ) /\ [_ ( m + ( M - 1 ) ) / k ]_ A e. CC ) -> ( ( k e. ( M ... N ) |-> A ) ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
123 |
95 120 122
|
syl2anc |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( k e. ( M ... N ) |-> A ) ` ( m + ( M - 1 ) ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
124 |
114 123
|
eqtrd |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( k e. ( M ... N ) |-> A ) ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) = [_ ( m + ( M - 1 ) ) / k ]_ A ) |
125 |
104 113 124
|
3eqtr4d |
|- ( ( ph /\ m e. ( 1 ... ( N + ( 1 - M ) ) ) ) -> ( ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ` m ) = ( ( k e. ( M ... N ) |-> A ) ` ( ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ` m ) ) ) |
126 |
5 19 83 85 125
|
fprod |
|- ( ph -> prod_ j e. ( M ... N ) ( ( k e. ( M ... N ) |-> A ) ` j ) = ( seq 1 ( x. , ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ) ` ( N + ( 1 - M ) ) ) ) |
127 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
128 |
19 127
|
eleqtrdi |
|- ( ph -> ( N + ( 1 - M ) ) e. ( ZZ>= ` 1 ) ) |
129 |
128 29 113
|
seqshft2 |
|- ( ph -> ( seq 1 ( x. , ( F o. ( n e. ( 1 ... ( N + ( 1 - M ) ) ) |-> ( n + ( M - 1 ) ) ) ) ) ` ( N + ( 1 - M ) ) ) = ( seq ( 1 + ( M - 1 ) ) ( x. , F ) ` ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) ) |
130 |
20
|
seqeq1d |
|- ( ph -> seq ( 1 + ( M - 1 ) ) ( x. , F ) = seq M ( x. , F ) ) |
131 |
130 21
|
fveq12d |
|- ( ph -> ( seq ( 1 + ( M - 1 ) ) ( x. , F ) ` ( ( N + ( 1 - M ) ) + ( M - 1 ) ) ) = ( seq M ( x. , F ) ` N ) ) |
132 |
126 129 131
|
3eqtrd |
|- ( ph -> prod_ j e. ( M ... N ) ( ( k e. ( M ... N ) |-> A ) ` j ) = ( seq M ( x. , F ) ` N ) ) |
133 |
4 132
|
eqtr3id |
|- ( ph -> prod_ k e. ( M ... N ) A = ( seq M ( x. , F ) ` N ) ) |