Step |
Hyp |
Ref |
Expression |
1 |
|
breprexp.n |
|- ( ph -> N e. NN0 ) |
2 |
|
breprexp.s |
|- ( ph -> S e. NN0 ) |
3 |
|
breprexp.z |
|- ( ph -> Z e. CC ) |
4 |
|
breprexp.h |
|- ( ph -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
5 |
|
breprexplemc.t |
|- ( ph -> T e. NN0 ) |
6 |
|
breprexplemc.s |
|- ( ph -> ( T + 1 ) <_ S ) |
7 |
|
breprexplemc.1 |
|- ( ph -> prod_ a e. ( 0 ..^ T ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = sum_ m e. ( 0 ... ( T x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) ) |
8 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
9 |
5 8
|
eleqtrdi |
|- ( ph -> T e. ( ZZ>= ` 0 ) ) |
10 |
|
fzosplitsn |
|- ( T e. ( ZZ>= ` 0 ) -> ( 0 ..^ ( T + 1 ) ) = ( ( 0 ..^ T ) u. { T } ) ) |
11 |
9 10
|
syl |
|- ( ph -> ( 0 ..^ ( T + 1 ) ) = ( ( 0 ..^ T ) u. { T } ) ) |
12 |
11
|
prodeq1d |
|- ( ph -> prod_ a e. ( 0 ..^ ( T + 1 ) ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = prod_ a e. ( ( 0 ..^ T ) u. { T } ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) ) |
13 |
|
nfv |
|- F/ a ph |
14 |
|
nfcv |
|- F/_ a sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) |
15 |
|
fzofi |
|- ( 0 ..^ T ) e. Fin |
16 |
15
|
a1i |
|- ( ph -> ( 0 ..^ T ) e. Fin ) |
17 |
|
fzonel |
|- -. T e. ( 0 ..^ T ) |
18 |
17
|
a1i |
|- ( ph -> -. T e. ( 0 ..^ T ) ) |
19 |
|
fzfid |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> ( 1 ... N ) e. Fin ) |
20 |
1
|
ad2antrr |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> N e. NN0 ) |
21 |
2
|
ad2antrr |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> S e. NN0 ) |
22 |
3
|
ad2antrr |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> Z e. CC ) |
23 |
4
|
adantr |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
24 |
23
|
adantr |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
25 |
5
|
nn0zd |
|- ( ph -> T e. ZZ ) |
26 |
2
|
nn0zd |
|- ( ph -> S e. ZZ ) |
27 |
5
|
nn0red |
|- ( ph -> T e. RR ) |
28 |
|
1red |
|- ( ph -> 1 e. RR ) |
29 |
27 28
|
readdcld |
|- ( ph -> ( T + 1 ) e. RR ) |
30 |
2
|
nn0red |
|- ( ph -> S e. RR ) |
31 |
27
|
lep1d |
|- ( ph -> T <_ ( T + 1 ) ) |
32 |
27 29 30 31 6
|
letrd |
|- ( ph -> T <_ S ) |
33 |
|
eluz1 |
|- ( T e. ZZ -> ( S e. ( ZZ>= ` T ) <-> ( S e. ZZ /\ T <_ S ) ) ) |
34 |
33
|
biimpar |
|- ( ( T e. ZZ /\ ( S e. ZZ /\ T <_ S ) ) -> S e. ( ZZ>= ` T ) ) |
35 |
25 26 32 34
|
syl12anc |
|- ( ph -> S e. ( ZZ>= ` T ) ) |
36 |
|
fzoss2 |
|- ( S e. ( ZZ>= ` T ) -> ( 0 ..^ T ) C_ ( 0 ..^ S ) ) |
37 |
35 36
|
syl |
|- ( ph -> ( 0 ..^ T ) C_ ( 0 ..^ S ) ) |
38 |
37
|
sselda |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ S ) ) |
39 |
38
|
adantr |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> a e. ( 0 ..^ S ) ) |
40 |
|
fz1ssnn |
|- ( 1 ... N ) C_ NN |
41 |
40
|
a1i |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> ( 1 ... N ) C_ NN ) |
42 |
41
|
sselda |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> b e. NN ) |
43 |
20 21 22 24 39 42
|
breprexplemb |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> ( ( L ` a ) ` b ) e. CC ) |
44 |
|
nnssnn0 |
|- NN C_ NN0 |
45 |
40 44
|
sstri |
|- ( 1 ... N ) C_ NN0 |
46 |
45
|
a1i |
|- ( ph -> ( 1 ... N ) C_ NN0 ) |
47 |
46
|
ralrimivw |
|- ( ph -> A. a e. ( 0 ..^ T ) ( 1 ... N ) C_ NN0 ) |
48 |
47
|
r19.21bi |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> ( 1 ... N ) C_ NN0 ) |
49 |
48
|
sselda |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> b e. NN0 ) |
50 |
22 49
|
expcld |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> ( Z ^ b ) e. CC ) |
51 |
43 50
|
mulcld |
|- ( ( ( ph /\ a e. ( 0 ..^ T ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) e. CC ) |
52 |
19 51
|
fsumcl |
|- ( ( ph /\ a e. ( 0 ..^ T ) ) -> sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) e. CC ) |
53 |
|
simpl |
|- ( ( a = T /\ b e. ( 1 ... N ) ) -> a = T ) |
54 |
53
|
fveq2d |
|- ( ( a = T /\ b e. ( 1 ... N ) ) -> ( L ` a ) = ( L ` T ) ) |
55 |
54
|
fveq1d |
|- ( ( a = T /\ b e. ( 1 ... N ) ) -> ( ( L ` a ) ` b ) = ( ( L ` T ) ` b ) ) |
56 |
55
|
oveq1d |
|- ( ( a = T /\ b e. ( 1 ... N ) ) -> ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) |
57 |
56
|
sumeq2dv |
|- ( a = T -> sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) |
58 |
|
fzfid |
|- ( ph -> ( 1 ... N ) e. Fin ) |
59 |
1
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> N e. NN0 ) |
60 |
2
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> S e. NN0 ) |
61 |
3
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> Z e. CC ) |
62 |
4
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
63 |
5
|
nn0ge0d |
|- ( ph -> 0 <_ T ) |
64 |
|
zltp1le |
|- ( ( T e. ZZ /\ S e. ZZ ) -> ( T < S <-> ( T + 1 ) <_ S ) ) |
65 |
25 26 64
|
syl2anc |
|- ( ph -> ( T < S <-> ( T + 1 ) <_ S ) ) |
66 |
6 65
|
mpbird |
|- ( ph -> T < S ) |
67 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
68 |
|
elfzo |
|- ( ( T e. ZZ /\ 0 e. ZZ /\ S e. ZZ ) -> ( T e. ( 0 ..^ S ) <-> ( 0 <_ T /\ T < S ) ) ) |
69 |
25 67 26 68
|
syl3anc |
|- ( ph -> ( T e. ( 0 ..^ S ) <-> ( 0 <_ T /\ T < S ) ) ) |
70 |
63 66 69
|
mpbir2and |
|- ( ph -> T e. ( 0 ..^ S ) ) |
71 |
70
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> T e. ( 0 ..^ S ) ) |
72 |
40
|
a1i |
|- ( ph -> ( 1 ... N ) C_ NN ) |
73 |
72
|
sselda |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> b e. NN ) |
74 |
59 60 61 62 71 73
|
breprexplemb |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( ( L ` T ) ` b ) e. CC ) |
75 |
46
|
sselda |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> b e. NN0 ) |
76 |
61 75
|
expcld |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( Z ^ b ) e. CC ) |
77 |
74 76
|
mulcld |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) e. CC ) |
78 |
58 77
|
fsumcl |
|- ( ph -> sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) e. CC ) |
79 |
13 14 16 5 18 52 57 78
|
fprodsplitsn |
|- ( ph -> prod_ a e. ( ( 0 ..^ T ) u. { T } ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = ( prod_ a e. ( 0 ..^ T ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) x. sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
80 |
7
|
oveq1d |
|- ( ph -> ( prod_ a e. ( 0 ..^ T ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) x. sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = ( sum_ m e. ( 0 ... ( T x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
81 |
|
fzfid |
|- ( ph -> ( 0 ... ( T x. N ) ) e. Fin ) |
82 |
40
|
a1i |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> ( 1 ... N ) C_ NN ) |
83 |
|
simpr |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> m e. ( 0 ... ( T x. N ) ) ) |
84 |
83
|
elfzelzd |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> m e. ZZ ) |
85 |
5
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> T e. NN0 ) |
86 |
58
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> ( 1 ... N ) e. Fin ) |
87 |
82 84 85 86
|
reprfi |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> ( ( 1 ... N ) ( repr ` T ) m ) e. Fin ) |
88 |
15
|
a1i |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( 0 ..^ T ) e. Fin ) |
89 |
1
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> N e. NN0 ) |
90 |
89
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> N e. NN0 ) |
91 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> S e. NN0 ) |
92 |
3
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> Z e. CC ) |
93 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
94 |
37
|
ad2antrr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( 0 ..^ T ) C_ ( 0 ..^ S ) ) |
95 |
94
|
sselda |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ S ) ) |
96 |
40
|
a1i |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( 1 ... N ) C_ NN ) |
97 |
84
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> m e. ZZ ) |
98 |
85
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> T e. NN0 ) |
99 |
|
simplr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> d e. ( ( 1 ... N ) ( repr ` T ) m ) ) |
100 |
96 97 98 99
|
reprf |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> d : ( 0 ..^ T ) --> ( 1 ... N ) ) |
101 |
|
simpr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ T ) ) |
102 |
100 101
|
ffvelrnd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. ( 1 ... N ) ) |
103 |
40 102
|
sselid |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. NN ) |
104 |
90 91 92 93 95 103
|
breprexplemb |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
105 |
88 104
|
fprodcl |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
106 |
3
|
ad2antrr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> Z e. CC ) |
107 |
|
fz0ssnn0 |
|- ( 0 ... ( T x. N ) ) C_ NN0 |
108 |
107 83
|
sselid |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> m e. NN0 ) |
109 |
108
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> m e. NN0 ) |
110 |
106 109
|
expcld |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( Z ^ m ) e. CC ) |
111 |
105 110
|
mulcld |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) e. CC ) |
112 |
87 111
|
fsumcl |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) e. CC ) |
113 |
81 58 112 77
|
fsum2mul |
|- ( ph -> sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = ( sum_ m e. ( 0 ... ( T x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
114 |
40
|
a1i |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( 1 ... N ) C_ NN ) |
115 |
|
simpr |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) |
116 |
115
|
elfzelzd |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> m e. ZZ ) |
117 |
116
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> m e. ZZ ) |
118 |
|
simpr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> b e. ( 1 ... N ) ) |
119 |
118
|
elfzelzd |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> b e. ZZ ) |
120 |
117 119
|
zsubcld |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( m - b ) e. ZZ ) |
121 |
5
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> T e. NN0 ) |
122 |
121
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> T e. NN0 ) |
123 |
58
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( 1 ... N ) e. Fin ) |
124 |
123
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( 1 ... N ) e. Fin ) |
125 |
114 120 122 124
|
reprfi |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) e. Fin ) |
126 |
74
|
adantlr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( L ` T ) ` b ) e. CC ) |
127 |
3
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> Z e. CC ) |
128 |
|
fz0ssnn0 |
|- ( 0 ... ( ( T + 1 ) x. N ) ) C_ NN0 |
129 |
128 115
|
sselid |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> m e. NN0 ) |
130 |
127 129
|
expcld |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( Z ^ m ) e. CC ) |
131 |
130
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( Z ^ m ) e. CC ) |
132 |
126 131
|
mulcld |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) e. CC ) |
133 |
15
|
a1i |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> ( 0 ..^ T ) e. Fin ) |
134 |
1
|
adantr |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> N e. NN0 ) |
135 |
134
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> N e. NN0 ) |
136 |
135
|
ad2antrr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> N e. NN0 ) |
137 |
2
|
ad4antr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> S e. NN0 ) |
138 |
127
|
ad3antrrr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> Z e. CC ) |
139 |
4
|
ad4antr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
140 |
38
|
ad5ant15 |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ S ) ) |
141 |
40
|
a1i |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> ( 1 ... N ) C_ NN ) |
142 |
120
|
ad2antrr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> ( m - b ) e. ZZ ) |
143 |
122
|
ad2antrr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> T e. NN0 ) |
144 |
|
simplr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) |
145 |
141 142 143 144
|
reprf |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> d : ( 0 ..^ T ) --> ( 1 ... N ) ) |
146 |
|
simpr |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ T ) ) |
147 |
145 146
|
ffvelrnd |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. ( 1 ... N ) ) |
148 |
40 147
|
sselid |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. NN ) |
149 |
136 137 138 139 140 148
|
breprexplemb |
|- ( ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) /\ a e. ( 0 ..^ T ) ) -> ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
150 |
133 149
|
fprodcl |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
151 |
125 132 150
|
fsummulc1 |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
152 |
151
|
sumeq2dv |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) = sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
153 |
|
elfzle2 |
|- ( m e. ( 0 ... ( ( T + 1 ) x. N ) ) -> m <_ ( ( T + 1 ) x. N ) ) |
154 |
153
|
adantl |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> m <_ ( ( T + 1 ) x. N ) ) |
155 |
134
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> N e. NN0 ) |
156 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> S e. NN0 ) |
157 |
127
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> Z e. CC ) |
158 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
159 |
25
|
peano2zd |
|- ( ph -> ( T + 1 ) e. ZZ ) |
160 |
|
eluz |
|- ( ( ( T + 1 ) e. ZZ /\ S e. ZZ ) -> ( S e. ( ZZ>= ` ( T + 1 ) ) <-> ( T + 1 ) <_ S ) ) |
161 |
160
|
biimpar |
|- ( ( ( ( T + 1 ) e. ZZ /\ S e. ZZ ) /\ ( T + 1 ) <_ S ) -> S e. ( ZZ>= ` ( T + 1 ) ) ) |
162 |
159 26 6 161
|
syl21anc |
|- ( ph -> S e. ( ZZ>= ` ( T + 1 ) ) ) |
163 |
|
fzoss2 |
|- ( S e. ( ZZ>= ` ( T + 1 ) ) -> ( 0 ..^ ( T + 1 ) ) C_ ( 0 ..^ S ) ) |
164 |
162 163
|
syl |
|- ( ph -> ( 0 ..^ ( T + 1 ) ) C_ ( 0 ..^ S ) ) |
165 |
164
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> ( 0 ..^ ( T + 1 ) ) C_ ( 0 ..^ S ) ) |
166 |
|
simplr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> x e. ( 0 ..^ ( T + 1 ) ) ) |
167 |
165 166
|
sseldd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> x e. ( 0 ..^ S ) ) |
168 |
|
simpr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> y e. NN ) |
169 |
155 156 157 158 167 168
|
breprexplemb |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ x e. ( 0 ..^ ( T + 1 ) ) ) /\ y e. NN ) -> ( ( L ` x ) ` y ) e. CC ) |
170 |
134 121 129 154 169
|
breprexplema |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) = sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) ) |
171 |
170
|
oveq1d |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = ( sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) ) |
172 |
126
|
adantr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> ( ( L ` T ) ` b ) e. CC ) |
173 |
150 172
|
mulcld |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) e. CC ) |
174 |
125 173
|
fsumcl |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) e. CC ) |
175 |
123 130 174
|
fsummulc1 |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) ) |
176 |
125 131 173
|
fsummulc1 |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) ) |
177 |
131
|
adantr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> ( Z ^ m ) e. CC ) |
178 |
150 172 177
|
mulassd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) -> ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
179 |
178
|
sumeq2dv |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
180 |
176 179
|
eqtrd |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
181 |
180
|
sumeq2dv |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ m ) ) = sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
182 |
171 175 181
|
3eqtrd |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = sum_ b e. ( 1 ... N ) sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
183 |
40
|
a1i |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( 1 ... N ) C_ NN ) |
184 |
|
1nn0 |
|- 1 e. NN0 |
185 |
184
|
a1i |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> 1 e. NN0 ) |
186 |
121 185
|
nn0addcld |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( T + 1 ) e. NN0 ) |
187 |
183 116 186 123
|
reprfi |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) e. Fin ) |
188 |
|
fzofi |
|- ( 0 ..^ ( T + 1 ) ) e. Fin |
189 |
188
|
a1i |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) -> ( 0 ..^ ( T + 1 ) ) e. Fin ) |
190 |
134
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> N e. NN0 ) |
191 |
2
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> S e. NN0 ) |
192 |
127
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> Z e. CC ) |
193 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
194 |
164
|
ad2antrr |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) -> ( 0 ..^ ( T + 1 ) ) C_ ( 0 ..^ S ) ) |
195 |
194
|
sselda |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> a e. ( 0 ..^ S ) ) |
196 |
40
|
a1i |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> ( 1 ... N ) C_ NN ) |
197 |
116
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> m e. ZZ ) |
198 |
186
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> ( T + 1 ) e. NN0 ) |
199 |
|
simplr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) |
200 |
196 197 198 199
|
reprf |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> d : ( 0 ..^ ( T + 1 ) ) --> ( 1 ... N ) ) |
201 |
|
simpr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> a e. ( 0 ..^ ( T + 1 ) ) ) |
202 |
200 201
|
ffvelrnd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> ( d ` a ) e. ( 1 ... N ) ) |
203 |
40 202
|
sselid |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> ( d ` a ) e. NN ) |
204 |
190 191 192 193 195 203
|
breprexplemb |
|- ( ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) /\ a e. ( 0 ..^ ( T + 1 ) ) ) -> ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
205 |
189 204
|
fprodcl |
|- ( ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ) -> prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
206 |
187 130 205
|
fsummulc1 |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) ) |
207 |
152 182 206
|
3eqtr2rd |
|- ( ( ph /\ m e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
208 |
207
|
sumeq2dv |
|- ( ph -> sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
209 |
|
oveq1 |
|- ( n = m -> ( n - b ) = ( m - b ) ) |
210 |
209
|
oveq2d |
|- ( n = m -> ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) = ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) ) |
211 |
210
|
sumeq1d |
|- ( n = m -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) ) |
212 |
|
oveq2 |
|- ( n = m -> ( Z ^ n ) = ( Z ^ m ) ) |
213 |
212
|
oveq2d |
|- ( n = m -> ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) |
214 |
211 213
|
oveq12d |
|- ( n = m -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
215 |
214
|
adantr |
|- ( ( n = m /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
216 |
215
|
sumeq2dv |
|- ( n = m -> sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) ) |
217 |
216
|
cbvsumv |
|- sum_ n e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( m - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) |
218 |
208 217
|
eqtr4di |
|- ( ph -> sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) = sum_ n e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) ) |
219 |
5 1
|
nn0mulcld |
|- ( ph -> ( T x. N ) e. NN0 ) |
220 |
|
oveq2 |
|- ( m = ( n - b ) -> ( ( 1 ... N ) ( repr ` T ) m ) = ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) ) |
221 |
220
|
sumeq1d |
|- ( m = ( n - b ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) ) |
222 |
|
oveq1 |
|- ( m = ( n - b ) -> ( m + b ) = ( ( n - b ) + b ) ) |
223 |
222
|
oveq2d |
|- ( m = ( n - b ) -> ( Z ^ ( m + b ) ) = ( Z ^ ( ( n - b ) + b ) ) ) |
224 |
223
|
oveq2d |
|- ( m = ( n - b ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) |
225 |
221 224
|
oveq12d |
|- ( m = ( n - b ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
226 |
40
|
a1i |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( 1 ... N ) C_ NN ) |
227 |
|
uzssz |
|- ( ZZ>= ` -u b ) C_ ZZ |
228 |
|
simp2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> m e. ( ZZ>= ` -u b ) ) |
229 |
227 228
|
sselid |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> m e. ZZ ) |
230 |
5
|
3ad2ant1 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> T e. NN0 ) |
231 |
58
|
3ad2ant1 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( 1 ... N ) e. Fin ) |
232 |
226 229 230 231
|
reprfi |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( ( 1 ... N ) ( repr ` T ) m ) e. Fin ) |
233 |
15
|
a1i |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( 0 ..^ T ) e. Fin ) |
234 |
59
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> N e. NN0 ) |
235 |
234
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> N e. NN0 ) |
236 |
60
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> S e. NN0 ) |
237 |
236
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> S e. NN0 ) |
238 |
61
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> Z e. CC ) |
239 |
238
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> Z e. CC ) |
240 |
62
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
241 |
240
|
ad2antrr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> L : ( 0 ..^ S ) --> ( CC ^m NN ) ) |
242 |
37
|
3ad2ant1 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( 0 ..^ T ) C_ ( 0 ..^ S ) ) |
243 |
242
|
adantr |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( 0 ..^ T ) C_ ( 0 ..^ S ) ) |
244 |
243
|
sselda |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ S ) ) |
245 |
40
|
a1i |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( 1 ... N ) C_ NN ) |
246 |
229
|
adantr |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> m e. ZZ ) |
247 |
230
|
adantr |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> T e. NN0 ) |
248 |
|
simpr |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> d e. ( ( 1 ... N ) ( repr ` T ) m ) ) |
249 |
245 246 247 248
|
reprf |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> d : ( 0 ..^ T ) --> ( 1 ... N ) ) |
250 |
249
|
adantr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> d : ( 0 ..^ T ) --> ( 1 ... N ) ) |
251 |
|
simpr |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> a e. ( 0 ..^ T ) ) |
252 |
250 251
|
ffvelrnd |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. ( 1 ... N ) ) |
253 |
40 252
|
sselid |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( d ` a ) e. NN ) |
254 |
235 237 239 241 244 253
|
breprexplemb |
|- ( ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) /\ a e. ( 0 ..^ T ) ) -> ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
255 |
233 254
|
fprodcl |
|- ( ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
256 |
232 255
|
fsumcl |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
257 |
71
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> T e. ( 0 ..^ S ) ) |
258 |
73
|
3adant2 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> b e. NN ) |
259 |
234 236 238 240 257 258
|
breprexplemb |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( ( L ` T ) ` b ) e. CC ) |
260 |
229
|
zcnd |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> m e. CC ) |
261 |
|
simp3 |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> b e. ( 1 ... N ) ) |
262 |
261
|
elfzelzd |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> b e. ZZ ) |
263 |
262
|
zcnd |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> b e. CC ) |
264 |
260 263
|
subnegd |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( m - -u b ) = ( m + b ) ) |
265 |
262
|
znegcld |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> -u b e. ZZ ) |
266 |
|
eluzle |
|- ( m e. ( ZZ>= ` -u b ) -> -u b <_ m ) |
267 |
228 266
|
syl |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> -u b <_ m ) |
268 |
|
znn0sub |
|- ( ( -u b e. ZZ /\ m e. ZZ ) -> ( -u b <_ m <-> ( m - -u b ) e. NN0 ) ) |
269 |
268
|
biimpa |
|- ( ( ( -u b e. ZZ /\ m e. ZZ ) /\ -u b <_ m ) -> ( m - -u b ) e. NN0 ) |
270 |
265 229 267 269
|
syl21anc |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( m - -u b ) e. NN0 ) |
271 |
264 270
|
eqeltrrd |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( m + b ) e. NN0 ) |
272 |
238 271
|
expcld |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( Z ^ ( m + b ) ) e. CC ) |
273 |
259 272
|
mulcld |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) e. CC ) |
274 |
256 273
|
mulcld |
|- ( ( ph /\ m e. ( ZZ>= ` -u b ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) e. CC ) |
275 |
59
|
adantr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> N e. NN0 ) |
276 |
|
ssidd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( 1 ... N ) C_ ( 1 ... N ) ) |
277 |
|
simpr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) |
278 |
277
|
elfzelzd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> n e. ZZ ) |
279 |
|
simplr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> b e. ( 1 ... N ) ) |
280 |
279
|
elfzelzd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> b e. ZZ ) |
281 |
278 280
|
zsubcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( n - b ) e. ZZ ) |
282 |
5
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> T e. NN0 ) |
283 |
27
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> T e. RR ) |
284 |
275
|
nn0red |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> N e. RR ) |
285 |
283 284
|
remulcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( T x. N ) e. RR ) |
286 |
280
|
zred |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> b e. RR ) |
287 |
219
|
adantr |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( T x. N ) e. NN0 ) |
288 |
287 75
|
nn0addcld |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( ( T x. N ) + b ) e. NN0 ) |
289 |
184
|
a1i |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> 1 e. NN0 ) |
290 |
288 289
|
nn0addcld |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( ( ( T x. N ) + b ) + 1 ) e. NN0 ) |
291 |
|
fz2ssnn0 |
|- ( ( ( ( T x. N ) + b ) + 1 ) e. NN0 -> ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) C_ NN0 ) |
292 |
290 291
|
syl |
|- ( ( ph /\ b e. ( 1 ... N ) ) -> ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) C_ NN0 ) |
293 |
292
|
sselda |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> n e. NN0 ) |
294 |
293
|
nn0red |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> n e. RR ) |
295 |
25
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> T e. ZZ ) |
296 |
275
|
nn0zd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> N e. ZZ ) |
297 |
295 296
|
zmulcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( T x. N ) e. ZZ ) |
298 |
297 280
|
zaddcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( T x. N ) + b ) e. ZZ ) |
299 |
|
elfzle1 |
|- ( n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) -> ( ( ( T x. N ) + b ) + 1 ) <_ n ) |
300 |
277 299
|
syl |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( ( T x. N ) + b ) + 1 ) <_ n ) |
301 |
|
zltp1le |
|- ( ( ( ( T x. N ) + b ) e. ZZ /\ n e. ZZ ) -> ( ( ( T x. N ) + b ) < n <-> ( ( ( T x. N ) + b ) + 1 ) <_ n ) ) |
302 |
301
|
biimpar |
|- ( ( ( ( ( T x. N ) + b ) e. ZZ /\ n e. ZZ ) /\ ( ( ( T x. N ) + b ) + 1 ) <_ n ) -> ( ( T x. N ) + b ) < n ) |
303 |
298 278 300 302
|
syl21anc |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( T x. N ) + b ) < n ) |
304 |
|
ltaddsub |
|- ( ( ( T x. N ) e. RR /\ b e. RR /\ n e. RR ) -> ( ( ( T x. N ) + b ) < n <-> ( T x. N ) < ( n - b ) ) ) |
305 |
304
|
biimpa |
|- ( ( ( ( T x. N ) e. RR /\ b e. RR /\ n e. RR ) /\ ( ( T x. N ) + b ) < n ) -> ( T x. N ) < ( n - b ) ) |
306 |
285 286 294 303 305
|
syl31anc |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( T x. N ) < ( n - b ) ) |
307 |
275 276 281 282 306
|
reprgt |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) = (/) ) |
308 |
307
|
sumeq1d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = sum_ d e. (/) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) ) |
309 |
|
sum0 |
|- sum_ d e. (/) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = 0 |
310 |
308 309
|
eqtrdi |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = 0 ) |
311 |
310
|
oveq1d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = ( 0 x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
312 |
74
|
adantr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( L ` T ) ` b ) e. CC ) |
313 |
61
|
adantr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> Z e. CC ) |
314 |
278
|
zcnd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> n e. CC ) |
315 |
280
|
zcnd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> b e. CC ) |
316 |
314 315
|
npcand |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( n - b ) + b ) = n ) |
317 |
316 293
|
eqeltrd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( n - b ) + b ) e. NN0 ) |
318 |
313 317
|
expcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( Z ^ ( ( n - b ) + b ) ) e. CC ) |
319 |
312 318
|
mulcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) e. CC ) |
320 |
319
|
mul02d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( 0 x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = 0 ) |
321 |
311 320
|
eqtrd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( ( ( ( T x. N ) + b ) + 1 ) ... ( ( T x. N ) + N ) ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = 0 ) |
322 |
40
|
a1i |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( 1 ... N ) C_ NN ) |
323 |
|
fzossfz |
|- ( 0 ..^ b ) C_ ( 0 ... b ) |
324 |
|
fzssz |
|- ( 0 ... b ) C_ ZZ |
325 |
323 324
|
sstri |
|- ( 0 ..^ b ) C_ ZZ |
326 |
|
simpr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n e. ( 0 ..^ b ) ) |
327 |
325 326
|
sselid |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n e. ZZ ) |
328 |
|
simplr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> b e. ( 1 ... N ) ) |
329 |
328
|
elfzelzd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> b e. ZZ ) |
330 |
327 329
|
zsubcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( n - b ) e. ZZ ) |
331 |
5
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> T e. NN0 ) |
332 |
330
|
zred |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( n - b ) e. RR ) |
333 |
|
0red |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> 0 e. RR ) |
334 |
27
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> T e. RR ) |
335 |
|
elfzolt2 |
|- ( n e. ( 0 ..^ b ) -> n < b ) |
336 |
335
|
adantl |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n < b ) |
337 |
327
|
zred |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n e. RR ) |
338 |
329
|
zred |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> b e. RR ) |
339 |
337 338
|
sublt0d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( n - b ) < 0 <-> n < b ) ) |
340 |
336 339
|
mpbird |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( n - b ) < 0 ) |
341 |
63
|
ad2antrr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> 0 <_ T ) |
342 |
332 333 334 340 341
|
ltletrd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( n - b ) < T ) |
343 |
322 330 331 342
|
reprlt |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) = (/) ) |
344 |
343
|
sumeq1d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = sum_ d e. (/) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) ) |
345 |
344 309
|
eqtrdi |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) = 0 ) |
346 |
345
|
oveq1d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = ( 0 x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
347 |
74
|
adantr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( L ` T ) ` b ) e. CC ) |
348 |
61
|
adantr |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> Z e. CC ) |
349 |
337
|
recnd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n e. CC ) |
350 |
338
|
recnd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> b e. CC ) |
351 |
349 350
|
npcand |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( n - b ) + b ) = n ) |
352 |
|
fzo0ssnn0 |
|- ( 0 ..^ b ) C_ NN0 |
353 |
352 326
|
sselid |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> n e. NN0 ) |
354 |
351 353
|
eqeltrd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( n - b ) + b ) e. NN0 ) |
355 |
348 354
|
expcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( Z ^ ( ( n - b ) + b ) ) e. CC ) |
356 |
347 355
|
mulcld |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) e. CC ) |
357 |
356
|
mul02d |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( 0 x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = 0 ) |
358 |
346 357
|
eqtrd |
|- ( ( ( ph /\ b e. ( 1 ... N ) ) /\ n e. ( 0 ..^ b ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) = 0 ) |
359 |
219 1 225 274 321 358
|
fsum2dsub |
|- ( ph -> sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = sum_ n e. ( 0 ... ( ( T x. N ) + N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
360 |
|
nn0sscn |
|- NN0 C_ CC |
361 |
360 5
|
sselid |
|- ( ph -> T e. CC ) |
362 |
360 1
|
sselid |
|- ( ph -> N e. CC ) |
363 |
361 362
|
adddirp1d |
|- ( ph -> ( ( T + 1 ) x. N ) = ( ( T x. N ) + N ) ) |
364 |
363
|
oveq2d |
|- ( ph -> ( 0 ... ( ( T + 1 ) x. N ) ) = ( 0 ... ( ( T x. N ) + N ) ) ) |
365 |
128 360
|
sstri |
|- ( 0 ... ( ( T + 1 ) x. N ) ) C_ CC |
366 |
|
simplr |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) |
367 |
365 366
|
sselid |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> n e. CC ) |
368 |
45 360
|
sstri |
|- ( 1 ... N ) C_ CC |
369 |
|
simpr |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> b e. ( 1 ... N ) ) |
370 |
368 369
|
sselid |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> b e. CC ) |
371 |
367 370
|
npcand |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( n - b ) + b ) = n ) |
372 |
371
|
eqcomd |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> n = ( ( n - b ) + b ) ) |
373 |
372
|
oveq2d |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( Z ^ n ) = ( Z ^ ( ( n - b ) + b ) ) ) |
374 |
373
|
oveq2d |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) |
375 |
374
|
oveq2d |
|- ( ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
376 |
375
|
sumeq2dv |
|- ( ( ph /\ n e. ( 0 ... ( ( T + 1 ) x. N ) ) ) -> sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
377 |
364 376
|
sumeq12dv |
|- ( ph -> sum_ n e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) = sum_ n e. ( 0 ... ( ( T x. N ) + N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( ( n - b ) + b ) ) ) ) ) |
378 |
359 377
|
eqtr4d |
|- ( ph -> sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = sum_ n e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) ( n - b ) ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ n ) ) ) ) |
379 |
105
|
adantlr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) e. CC ) |
380 |
110
|
adantlr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( Z ^ m ) e. CC ) |
381 |
77
|
adantlr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) e. CC ) |
382 |
381
|
adantr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) e. CC ) |
383 |
379 380 382
|
mulassd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( Z ^ m ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) ) |
384 |
74
|
ad4ant13 |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( L ` T ) ` b ) e. CC ) |
385 |
76
|
ad4ant13 |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( Z ^ b ) e. CC ) |
386 |
380 384 385
|
mulassd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( Z ^ m ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ b ) ) = ( ( Z ^ m ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
387 |
384 380 385
|
mulassd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) x. ( Z ^ b ) ) = ( ( ( L ` T ) ` b ) x. ( ( Z ^ m ) x. ( Z ^ b ) ) ) ) |
388 |
380 384
|
mulcomd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( Z ^ m ) x. ( ( L ` T ) ` b ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) ) |
389 |
388
|
oveq1d |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( Z ^ m ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ b ) ) = ( ( ( ( L ` T ) ` b ) x. ( Z ^ m ) ) x. ( Z ^ b ) ) ) |
390 |
106
|
adantlr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> Z e. CC ) |
391 |
75
|
ad4ant13 |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> b e. NN0 ) |
392 |
109
|
adantlr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> m e. NN0 ) |
393 |
390 391 392
|
expaddd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( Z ^ ( m + b ) ) = ( ( Z ^ m ) x. ( Z ^ b ) ) ) |
394 |
393
|
oveq2d |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) = ( ( ( L ` T ) ` b ) x. ( ( Z ^ m ) x. ( Z ^ b ) ) ) ) |
395 |
387 389 394
|
3eqtr4d |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( ( Z ^ m ) x. ( ( L ` T ) ` b ) ) x. ( Z ^ b ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) |
396 |
386 395
|
eqtr3d |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( Z ^ m ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) |
397 |
396
|
oveq2d |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( Z ^ m ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) = ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) ) |
398 |
383 397
|
eqtrd |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) ) |
399 |
398
|
sumeq2dv |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) ) |
400 |
87
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( 1 ... N ) ( repr ` T ) m ) e. Fin ) |
401 |
111
|
adantlr |
|- ( ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) /\ d e. ( ( 1 ... N ) ( repr ` T ) m ) ) -> ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) e. CC ) |
402 |
400 381 401
|
fsummulc1 |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
403 |
74
|
adantlr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( L ` T ) ` b ) e. CC ) |
404 |
61
|
adantlr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> Z e. CC ) |
405 |
108
|
adantr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> m e. NN0 ) |
406 |
75
|
adantlr |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> b e. NN0 ) |
407 |
405 406
|
nn0addcld |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( m + b ) e. NN0 ) |
408 |
404 407
|
expcld |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( Z ^ ( m + b ) ) e. CC ) |
409 |
403 408
|
mulcld |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) e. CC ) |
410 |
400 409 379
|
fsummulc1 |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) ) |
411 |
399 402 410
|
3eqtr4rd |
|- ( ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) /\ b e. ( 1 ... N ) ) -> ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
412 |
411
|
sumeq2dv |
|- ( ( ph /\ m e. ( 0 ... ( T x. N ) ) ) -> sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
413 |
412
|
sumeq2dv |
|- ( ph -> sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ ( m + b ) ) ) ) = sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) ) |
414 |
218 378 413
|
3eqtr2rd |
|- ( ph -> sum_ m e. ( 0 ... ( T x. N ) ) sum_ b e. ( 1 ... N ) ( sum_ d e. ( ( 1 ... N ) ( repr ` T ) m ) ( prod_ a e. ( 0 ..^ T ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) x. ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) ) |
415 |
80 113 414
|
3eqtr2d |
|- ( ph -> ( prod_ a e. ( 0 ..^ T ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) x. sum_ b e. ( 1 ... N ) ( ( ( L ` T ) ` b ) x. ( Z ^ b ) ) ) = sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) ) |
416 |
12 79 415
|
3eqtrd |
|- ( ph -> prod_ a e. ( 0 ..^ ( T + 1 ) ) sum_ b e. ( 1 ... N ) ( ( ( L ` a ) ` b ) x. ( Z ^ b ) ) = sum_ m e. ( 0 ... ( ( T + 1 ) x. N ) ) sum_ d e. ( ( 1 ... N ) ( repr ` ( T + 1 ) ) m ) ( prod_ a e. ( 0 ..^ ( T + 1 ) ) ( ( L ` a ) ` ( d ` a ) ) x. ( Z ^ m ) ) ) |