Step |
Hyp |
Ref |
Expression |
1 |
|
dvply1.f |
|- ( ph -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) |
2 |
|
dvply1.g |
|- ( ph -> G = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) ) |
3 |
|
dvply1.a |
|- ( ph -> A : NN0 --> CC ) |
4 |
|
dvply1.b |
|- B = ( k e. NN0 |-> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) ) |
5 |
|
dvply1.n |
|- ( ph -> N e. NN0 ) |
6 |
1
|
oveq2d |
|- ( ph -> ( CC _D F ) = ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) ) |
7 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
8 |
7
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
9 |
8
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
10 |
|
cnelprrecn |
|- CC e. { RR , CC } |
11 |
10
|
a1i |
|- ( ph -> CC e. { RR , CC } ) |
12 |
7
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
13 |
|
unicntop |
|- CC = U. ( TopOpen ` CCfld ) |
14 |
13
|
topopn |
|- ( ( TopOpen ` CCfld ) e. Top -> CC e. ( TopOpen ` CCfld ) ) |
15 |
12 14
|
mp1i |
|- ( ph -> CC e. ( TopOpen ` CCfld ) ) |
16 |
|
fzfid |
|- ( ph -> ( 0 ... N ) e. Fin ) |
17 |
|
elfznn0 |
|- ( k e. ( 0 ... N ) -> k e. NN0 ) |
18 |
|
ffvelrn |
|- ( ( A : NN0 --> CC /\ k e. NN0 ) -> ( A ` k ) e. CC ) |
19 |
3 17 18
|
syl2an |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC ) |
20 |
19
|
adantr |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( A ` k ) e. CC ) |
21 |
|
simpr |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> z e. CC ) |
22 |
17
|
ad2antlr |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> k e. NN0 ) |
23 |
21 22
|
expcld |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( z ^ k ) e. CC ) |
24 |
20 23
|
mulcld |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC ) |
25 |
24
|
3impa |
|- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. ( z ^ k ) ) e. CC ) |
26 |
19
|
3adant3 |
|- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( A ` k ) e. CC ) |
27 |
|
0cnd |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ k = 0 ) -> 0 e. CC ) |
28 |
|
simpl2 |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. ( 0 ... N ) ) |
29 |
28 17
|
syl |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN0 ) |
30 |
29
|
nn0cnd |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. CC ) |
31 |
|
simpl3 |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> z e. CC ) |
32 |
|
simpr |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> -. k = 0 ) |
33 |
|
elnn0 |
|- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) ) |
34 |
29 33
|
sylib |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k e. NN \/ k = 0 ) ) |
35 |
|
orel2 |
|- ( -. k = 0 -> ( ( k e. NN \/ k = 0 ) -> k e. NN ) ) |
36 |
32 34 35
|
sylc |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> k e. NN ) |
37 |
|
nnm1nn0 |
|- ( k e. NN -> ( k - 1 ) e. NN0 ) |
38 |
36 37
|
syl |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k - 1 ) e. NN0 ) |
39 |
31 38
|
expcld |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( z ^ ( k - 1 ) ) e. CC ) |
40 |
30 39
|
mulcld |
|- ( ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) /\ -. k = 0 ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC ) |
41 |
27 40
|
ifclda |
|- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC ) |
42 |
26 41
|
mulcld |
|- ( ( ph /\ k e. ( 0 ... N ) /\ z e. CC ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC ) |
43 |
10
|
a1i |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> CC e. { RR , CC } ) |
44 |
|
c0ex |
|- 0 e. _V |
45 |
|
ovex |
|- ( k x. ( z ^ ( k - 1 ) ) ) e. _V |
46 |
44 45
|
ifex |
|- if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V |
47 |
46
|
a1i |
|- ( ( ( ph /\ k e. ( 0 ... N ) ) /\ z e. CC ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. _V ) |
48 |
17
|
adantl |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> k e. NN0 ) |
49 |
|
dvexp2 |
|- ( k e. NN0 -> ( CC _D ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) |
50 |
48 49
|
syl |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( CC _D ( z e. CC |-> ( z ^ k ) ) ) = ( z e. CC |-> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) |
51 |
43 23 47 50 19
|
dvmptcmul |
|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( CC _D ( z e. CC |-> ( ( A ` k ) x. ( z ^ k ) ) ) ) = ( z e. CC |-> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) ) |
52 |
9 7 11 15 16 25 42 51
|
dvmptfsum |
|- ( ph -> ( CC _D ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) ) |
53 |
|
elfznn |
|- ( k e. ( 1 ... N ) -> k e. NN ) |
54 |
53
|
nnne0d |
|- ( k e. ( 1 ... N ) -> k =/= 0 ) |
55 |
54
|
neneqd |
|- ( k e. ( 1 ... N ) -> -. k = 0 ) |
56 |
55
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 ) |
57 |
56
|
iffalsed |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) ) |
58 |
57
|
oveq2d |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) ) |
59 |
58
|
sumeq2dv |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) ) |
60 |
|
1eluzge0 |
|- 1 e. ( ZZ>= ` 0 ) |
61 |
|
fzss1 |
|- ( 1 e. ( ZZ>= ` 0 ) -> ( 1 ... N ) C_ ( 0 ... N ) ) |
62 |
60 61
|
mp1i |
|- ( ( ph /\ z e. CC ) -> ( 1 ... N ) C_ ( 0 ... N ) ) |
63 |
3
|
adantr |
|- ( ( ph /\ z e. CC ) -> A : NN0 --> CC ) |
64 |
53
|
nnnn0d |
|- ( k e. ( 1 ... N ) -> k e. NN0 ) |
65 |
63 64 18
|
syl2an |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( A ` k ) e. CC ) |
66 |
54
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k =/= 0 ) |
67 |
66
|
neneqd |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> -. k = 0 ) |
68 |
67
|
iffalsed |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = ( k x. ( z ^ ( k - 1 ) ) ) ) |
69 |
64
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. NN0 ) |
70 |
69
|
nn0cnd |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> k e. CC ) |
71 |
|
simplr |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> z e. CC ) |
72 |
53 37
|
syl |
|- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 ) |
73 |
72
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 ) |
74 |
71 73
|
expcld |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( z ^ ( k - 1 ) ) e. CC ) |
75 |
70 74
|
mulcld |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( k x. ( z ^ ( k - 1 ) ) ) e. CC ) |
76 |
68 75
|
eqeltrd |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC ) |
77 |
65 76
|
mulcld |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) e. CC ) |
78 |
|
eldifn |
|- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( 1 ... N ) ) |
79 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
80 |
79
|
oveq1i |
|- ( ( 0 + 1 ) ... N ) = ( 1 ... N ) |
81 |
80
|
eleq2i |
|- ( k e. ( ( 0 + 1 ) ... N ) <-> k e. ( 1 ... N ) ) |
82 |
78 81
|
sylnibr |
|- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> -. k e. ( ( 0 + 1 ) ... N ) ) |
83 |
82
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> -. k e. ( ( 0 + 1 ) ... N ) ) |
84 |
|
eldifi |
|- ( k e. ( ( 0 ... N ) \ ( 1 ... N ) ) -> k e. ( 0 ... N ) ) |
85 |
84
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k e. ( 0 ... N ) ) |
86 |
|
nn0uz |
|- NN0 = ( ZZ>= ` 0 ) |
87 |
5 86
|
eleqtrdi |
|- ( ph -> N e. ( ZZ>= ` 0 ) ) |
88 |
87
|
ad2antrr |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> N e. ( ZZ>= ` 0 ) ) |
89 |
|
elfzp12 |
|- ( N e. ( ZZ>= ` 0 ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) ) |
90 |
88 89
|
syl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k e. ( 0 ... N ) <-> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) ) |
91 |
85 90
|
mpbid |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) ) |
92 |
|
orel2 |
|- ( -. k e. ( ( 0 + 1 ) ... N ) -> ( ( k = 0 \/ k e. ( ( 0 + 1 ) ... N ) ) -> k = 0 ) ) |
93 |
83 91 92
|
sylc |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> k = 0 ) |
94 |
93
|
iftrued |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) = 0 ) |
95 |
94
|
oveq2d |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = ( ( A ` k ) x. 0 ) ) |
96 |
63 17 18
|
syl2an |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( A ` k ) e. CC ) |
97 |
96
|
mul01d |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( A ` k ) x. 0 ) = 0 ) |
98 |
84 97
|
sylan2 |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. 0 ) = 0 ) |
99 |
95 98
|
eqtrd |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( ( 0 ... N ) \ ( 1 ... N ) ) ) -> ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = 0 ) |
100 |
|
fzfid |
|- ( ( ph /\ z e. CC ) -> ( 0 ... N ) e. Fin ) |
101 |
62 77 99 100
|
fsumss |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) |
102 |
|
elfznn0 |
|- ( j e. ( 0 ... ( N - 1 ) ) -> j e. NN0 ) |
103 |
102
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. NN0 ) |
104 |
103
|
nn0cnd |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> j e. CC ) |
105 |
|
ax-1cn |
|- 1 e. CC |
106 |
|
pncan |
|- ( ( j e. CC /\ 1 e. CC ) -> ( ( j + 1 ) - 1 ) = j ) |
107 |
104 105 106
|
sylancl |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) - 1 ) = j ) |
108 |
107
|
oveq2d |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ ( ( j + 1 ) - 1 ) ) = ( z ^ j ) ) |
109 |
108
|
oveq2d |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ j ) ) ) |
110 |
109
|
oveq2d |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ j ) ) ) ) |
111 |
3
|
ad2antrr |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> A : NN0 --> CC ) |
112 |
|
peano2nn0 |
|- ( j e. NN0 -> ( j + 1 ) e. NN0 ) |
113 |
102 112
|
syl |
|- ( j e. ( 0 ... ( N - 1 ) ) -> ( j + 1 ) e. NN0 ) |
114 |
113
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. NN0 ) |
115 |
111 114
|
ffvelrnd |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( A ` ( j + 1 ) ) e. CC ) |
116 |
114
|
nn0cnd |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( j + 1 ) e. CC ) |
117 |
|
simplr |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> z e. CC ) |
118 |
117 103
|
expcld |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( z ^ j ) e. CC ) |
119 |
115 116 118
|
mulassd |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) x. ( z ^ j ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ j ) ) ) ) |
120 |
115 116
|
mulcomd |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) ) |
121 |
120
|
oveq1d |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( ( A ` ( j + 1 ) ) x. ( j + 1 ) ) x. ( z ^ j ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) ) |
122 |
110 119 121
|
3eqtr2d |
|- ( ( ( ph /\ z e. CC ) /\ j e. ( 0 ... ( N - 1 ) ) ) -> ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) ) |
123 |
122
|
sumeq2dv |
|- ( ( ph /\ z e. CC ) -> sum_ j e. ( 0 ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) ) |
124 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
125 |
124
|
oveq1i |
|- ( ( 1 - 1 ) ... ( N - 1 ) ) = ( 0 ... ( N - 1 ) ) |
126 |
125
|
sumeq1i |
|- sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) |
127 |
|
oveq1 |
|- ( k = j -> ( k + 1 ) = ( j + 1 ) ) |
128 |
|
fvoveq1 |
|- ( k = j -> ( A ` ( k + 1 ) ) = ( A ` ( j + 1 ) ) ) |
129 |
127 128
|
oveq12d |
|- ( k = j -> ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) = ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) ) |
130 |
|
oveq2 |
|- ( k = j -> ( z ^ k ) = ( z ^ j ) ) |
131 |
129 130
|
oveq12d |
|- ( k = j -> ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) ) |
132 |
131
|
cbvsumv |
|- sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) = sum_ j e. ( 0 ... ( N - 1 ) ) ( ( ( j + 1 ) x. ( A ` ( j + 1 ) ) ) x. ( z ^ j ) ) |
133 |
123 126 132
|
3eqtr4g |
|- ( ( ph /\ z e. CC ) -> sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) ) |
134 |
|
1zzd |
|- ( ( ph /\ z e. CC ) -> 1 e. ZZ ) |
135 |
5
|
adantr |
|- ( ( ph /\ z e. CC ) -> N e. NN0 ) |
136 |
135
|
nn0zd |
|- ( ( ph /\ z e. CC ) -> N e. ZZ ) |
137 |
65 75
|
mulcld |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 1 ... N ) ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) e. CC ) |
138 |
|
fveq2 |
|- ( k = ( j + 1 ) -> ( A ` k ) = ( A ` ( j + 1 ) ) ) |
139 |
|
id |
|- ( k = ( j + 1 ) -> k = ( j + 1 ) ) |
140 |
|
oveq1 |
|- ( k = ( j + 1 ) -> ( k - 1 ) = ( ( j + 1 ) - 1 ) ) |
141 |
140
|
oveq2d |
|- ( k = ( j + 1 ) -> ( z ^ ( k - 1 ) ) = ( z ^ ( ( j + 1 ) - 1 ) ) ) |
142 |
139 141
|
oveq12d |
|- ( k = ( j + 1 ) -> ( k x. ( z ^ ( k - 1 ) ) ) = ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) |
143 |
138 142
|
oveq12d |
|- ( k = ( j + 1 ) -> ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) ) |
144 |
134 134 136 137 143
|
fsumshftm |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = sum_ j e. ( ( 1 - 1 ) ... ( N - 1 ) ) ( ( A ` ( j + 1 ) ) x. ( ( j + 1 ) x. ( z ^ ( ( j + 1 ) - 1 ) ) ) ) ) |
145 |
|
elfznn0 |
|- ( k e. ( 0 ... ( N - 1 ) ) -> k e. NN0 ) |
146 |
145
|
adantl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> k e. NN0 ) |
147 |
|
ovex |
|- ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V |
148 |
4
|
fvmpt2 |
|- ( ( k e. NN0 /\ ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) e. _V ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) ) |
149 |
146 147 148
|
sylancl |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( B ` k ) = ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) ) |
150 |
149
|
oveq1d |
|- ( ( ( ph /\ z e. CC ) /\ k e. ( 0 ... ( N - 1 ) ) ) -> ( ( B ` k ) x. ( z ^ k ) ) = ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) ) |
151 |
150
|
sumeq2dv |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( ( k + 1 ) x. ( A ` ( k + 1 ) ) ) x. ( z ^ k ) ) ) |
152 |
133 144 151
|
3eqtr4d |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 1 ... N ) ( ( A ` k ) x. ( k x. ( z ^ ( k - 1 ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) |
153 |
59 101 152
|
3eqtr3d |
|- ( ( ph /\ z e. CC ) -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) = sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) |
154 |
153
|
mpteq2dva |
|- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... ( N - 1 ) ) ( ( B ` k ) x. ( z ^ k ) ) ) ) |
155 |
154 2
|
eqtr4d |
|- ( ph -> ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. if ( k = 0 , 0 , ( k x. ( z ^ ( k - 1 ) ) ) ) ) ) = G ) |
156 |
6 52 155
|
3eqtrd |
|- ( ph -> ( CC _D F ) = G ) |