Step |
Hyp |
Ref |
Expression |
1 |
|
elfzelz |
|- ( k e. ( 0 ... N ) -> k e. ZZ ) |
2 |
|
bcpascm1 |
|- ( ( N e. NN /\ k e. ZZ ) -> ( ( ( N - 1 ) _C k ) + ( ( N - 1 ) _C ( k - 1 ) ) ) = ( N _C k ) ) |
3 |
1 2
|
sylan2 |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( ( N - 1 ) _C k ) + ( ( N - 1 ) _C ( k - 1 ) ) ) = ( N _C k ) ) |
4 |
3
|
eqcomd |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( N _C k ) = ( ( ( N - 1 ) _C k ) + ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
5 |
4
|
oveq2d |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( -u 1 ^ k ) x. ( N _C k ) ) = ( ( -u 1 ^ k ) x. ( ( ( N - 1 ) _C k ) + ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
6 |
|
ax-1cn |
|- 1 e. CC |
7 |
|
negcl |
|- ( 1 e. CC -> -u 1 e. CC ) |
8 |
|
elfznn0 |
|- ( k e. ( 0 ... N ) -> k e. NN0 ) |
9 |
|
expcl |
|- ( ( -u 1 e. CC /\ k e. NN0 ) -> ( -u 1 ^ k ) e. CC ) |
10 |
7 8 9
|
syl2an |
|- ( ( 1 e. CC /\ k e. ( 0 ... N ) ) -> ( -u 1 ^ k ) e. CC ) |
11 |
6 10
|
mpan |
|- ( k e. ( 0 ... N ) -> ( -u 1 ^ k ) e. CC ) |
12 |
11
|
adantl |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( -u 1 ^ k ) e. CC ) |
13 |
|
nnm1nn0 |
|- ( N e. NN -> ( N - 1 ) e. NN0 ) |
14 |
|
bccl |
|- ( ( ( N - 1 ) e. NN0 /\ k e. ZZ ) -> ( ( N - 1 ) _C k ) e. NN0 ) |
15 |
14
|
nn0cnd |
|- ( ( ( N - 1 ) e. NN0 /\ k e. ZZ ) -> ( ( N - 1 ) _C k ) e. CC ) |
16 |
13 1 15
|
syl2an |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( N - 1 ) _C k ) e. CC ) |
17 |
|
peano2zm |
|- ( k e. ZZ -> ( k - 1 ) e. ZZ ) |
18 |
1 17
|
syl |
|- ( k e. ( 0 ... N ) -> ( k - 1 ) e. ZZ ) |
19 |
|
bccl |
|- ( ( ( N - 1 ) e. NN0 /\ ( k - 1 ) e. ZZ ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. NN0 ) |
20 |
19
|
nn0cnd |
|- ( ( ( N - 1 ) e. NN0 /\ ( k - 1 ) e. ZZ ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. CC ) |
21 |
13 18 20
|
syl2an |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. CC ) |
22 |
12 16 21
|
adddid |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( -u 1 ^ k ) x. ( ( ( N - 1 ) _C k ) + ( ( N - 1 ) _C ( k - 1 ) ) ) ) = ( ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
23 |
5 22
|
eqtrd |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( -u 1 ^ k ) x. ( N _C k ) ) = ( ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
24 |
23
|
sumeq2dv |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( N _C k ) ) = sum_ k e. ( 0 ... N ) ( ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
25 |
|
fzfid |
|- ( N e. NN -> ( 0 ... N ) e. Fin ) |
26 |
|
neg1cn |
|- -u 1 e. CC |
27 |
26
|
a1i |
|- ( N e. NN -> -u 1 e. CC ) |
28 |
27 8 9
|
syl2an |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( -u 1 ^ k ) e. CC ) |
29 |
28 16
|
mulcld |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) e. CC ) |
30 |
|
1z |
|- 1 e. ZZ |
31 |
30
|
a1i |
|- ( k e. ( 0 ... N ) -> 1 e. ZZ ) |
32 |
1 31
|
zsubcld |
|- ( k e. ( 0 ... N ) -> ( k - 1 ) e. ZZ ) |
33 |
13 32 20
|
syl2an |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. CC ) |
34 |
28 33
|
mulcld |
|- ( ( N e. NN /\ k e. ( 0 ... N ) ) -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) e. CC ) |
35 |
25 29 34
|
fsumadd |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = ( sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
36 |
30
|
a1i |
|- ( N e. NN -> 1 e. ZZ ) |
37 |
|
0zd |
|- ( N e. NN -> 0 e. ZZ ) |
38 |
|
nnz |
|- ( N e. NN -> N e. ZZ ) |
39 |
|
oveq2 |
|- ( k = ( j - 1 ) -> ( -u 1 ^ k ) = ( -u 1 ^ ( j - 1 ) ) ) |
40 |
|
oveq2 |
|- ( k = ( j - 1 ) -> ( ( N - 1 ) _C k ) = ( ( N - 1 ) _C ( j - 1 ) ) ) |
41 |
39 40
|
oveq12d |
|- ( k = ( j - 1 ) -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) = ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) ) |
42 |
36 37 38 29 41
|
fsumshft |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) = sum_ j e. ( ( 0 + 1 ) ... ( N + 1 ) ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) ) |
43 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
44 |
43
|
oveq1i |
|- ( ( 0 + 1 ) ... ( N + 1 ) ) = ( 1 ... ( N + 1 ) ) |
45 |
44
|
a1i |
|- ( N e. NN -> ( ( 0 + 1 ) ... ( N + 1 ) ) = ( 1 ... ( N + 1 ) ) ) |
46 |
45
|
sumeq1d |
|- ( N e. NN -> sum_ j e. ( ( 0 + 1 ) ... ( N + 1 ) ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = sum_ j e. ( 1 ... ( N + 1 ) ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) ) |
47 |
|
elnnuz |
|- ( N e. NN <-> N e. ( ZZ>= ` 1 ) ) |
48 |
47
|
biimpi |
|- ( N e. NN -> N e. ( ZZ>= ` 1 ) ) |
49 |
26
|
a1i |
|- ( ( N e. NN /\ j e. ( 1 ... ( N + 1 ) ) ) -> -u 1 e. CC ) |
50 |
|
elfznn |
|- ( j e. ( 1 ... ( N + 1 ) ) -> j e. NN ) |
51 |
|
nnm1nn0 |
|- ( j e. NN -> ( j - 1 ) e. NN0 ) |
52 |
50 51
|
syl |
|- ( j e. ( 1 ... ( N + 1 ) ) -> ( j - 1 ) e. NN0 ) |
53 |
52
|
adantl |
|- ( ( N e. NN /\ j e. ( 1 ... ( N + 1 ) ) ) -> ( j - 1 ) e. NN0 ) |
54 |
49 53
|
expcld |
|- ( ( N e. NN /\ j e. ( 1 ... ( N + 1 ) ) ) -> ( -u 1 ^ ( j - 1 ) ) e. CC ) |
55 |
|
elfzelz |
|- ( j e. ( 1 ... ( N + 1 ) ) -> j e. ZZ ) |
56 |
|
elfzel1 |
|- ( j e. ( 1 ... ( N + 1 ) ) -> 1 e. ZZ ) |
57 |
55 56
|
zsubcld |
|- ( j e. ( 1 ... ( N + 1 ) ) -> ( j - 1 ) e. ZZ ) |
58 |
|
bccl |
|- ( ( ( N - 1 ) e. NN0 /\ ( j - 1 ) e. ZZ ) -> ( ( N - 1 ) _C ( j - 1 ) ) e. NN0 ) |
59 |
58
|
nn0cnd |
|- ( ( ( N - 1 ) e. NN0 /\ ( j - 1 ) e. ZZ ) -> ( ( N - 1 ) _C ( j - 1 ) ) e. CC ) |
60 |
13 57 59
|
syl2an |
|- ( ( N e. NN /\ j e. ( 1 ... ( N + 1 ) ) ) -> ( ( N - 1 ) _C ( j - 1 ) ) e. CC ) |
61 |
54 60
|
mulcld |
|- ( ( N e. NN /\ j e. ( 1 ... ( N + 1 ) ) ) -> ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) e. CC ) |
62 |
|
oveq1 |
|- ( j = ( N + 1 ) -> ( j - 1 ) = ( ( N + 1 ) - 1 ) ) |
63 |
62
|
oveq2d |
|- ( j = ( N + 1 ) -> ( -u 1 ^ ( j - 1 ) ) = ( -u 1 ^ ( ( N + 1 ) - 1 ) ) ) |
64 |
62
|
oveq2d |
|- ( j = ( N + 1 ) -> ( ( N - 1 ) _C ( j - 1 ) ) = ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) |
65 |
63 64
|
oveq12d |
|- ( j = ( N + 1 ) -> ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) ) |
66 |
48 61 65
|
fsump1 |
|- ( N e. NN -> sum_ j e. ( 1 ... ( N + 1 ) ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = ( sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) + ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) ) ) |
67 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
68 |
|
pncan1 |
|- ( N e. CC -> ( ( N + 1 ) - 1 ) = N ) |
69 |
67 68
|
syl |
|- ( N e. NN -> ( ( N + 1 ) - 1 ) = N ) |
70 |
|
nnnn0 |
|- ( N e. NN -> N e. NN0 ) |
71 |
69 70
|
eqeltrd |
|- ( N e. NN -> ( ( N + 1 ) - 1 ) e. NN0 ) |
72 |
71
|
nn0zd |
|- ( N e. NN -> ( ( N + 1 ) - 1 ) e. ZZ ) |
73 |
|
nnre |
|- ( N e. NN -> N e. RR ) |
74 |
|
ltm1 |
|- ( N e. RR -> ( N - 1 ) < N ) |
75 |
73 74
|
syl |
|- ( N e. NN -> ( N - 1 ) < N ) |
76 |
75 69
|
breqtrrd |
|- ( N e. NN -> ( N - 1 ) < ( ( N + 1 ) - 1 ) ) |
77 |
76
|
olcd |
|- ( N e. NN -> ( ( ( N + 1 ) - 1 ) < 0 \/ ( N - 1 ) < ( ( N + 1 ) - 1 ) ) ) |
78 |
|
bcval4 |
|- ( ( ( N - 1 ) e. NN0 /\ ( ( N + 1 ) - 1 ) e. ZZ /\ ( ( ( N + 1 ) - 1 ) < 0 \/ ( N - 1 ) < ( ( N + 1 ) - 1 ) ) ) -> ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) = 0 ) |
79 |
13 72 77 78
|
syl3anc |
|- ( N e. NN -> ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) = 0 ) |
80 |
79
|
oveq2d |
|- ( N e. NN -> ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) = ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. 0 ) ) |
81 |
27 71
|
expcld |
|- ( N e. NN -> ( -u 1 ^ ( ( N + 1 ) - 1 ) ) e. CC ) |
82 |
81
|
mul01d |
|- ( N e. NN -> ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. 0 ) = 0 ) |
83 |
80 82
|
eqtrd |
|- ( N e. NN -> ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) = 0 ) |
84 |
83
|
oveq2d |
|- ( N e. NN -> ( sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) + ( ( -u 1 ^ ( ( N + 1 ) - 1 ) ) x. ( ( N - 1 ) _C ( ( N + 1 ) - 1 ) ) ) ) = ( sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) + 0 ) ) |
85 |
|
oveq1 |
|- ( j = k -> ( j - 1 ) = ( k - 1 ) ) |
86 |
85
|
oveq2d |
|- ( j = k -> ( -u 1 ^ ( j - 1 ) ) = ( -u 1 ^ ( k - 1 ) ) ) |
87 |
85
|
oveq2d |
|- ( j = k -> ( ( N - 1 ) _C ( j - 1 ) ) = ( ( N - 1 ) _C ( k - 1 ) ) ) |
88 |
86 87
|
oveq12d |
|- ( j = k -> ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
89 |
88
|
cbvsumv |
|- sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) |
90 |
89
|
a1i |
|- ( N e. NN -> sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
91 |
90
|
oveq1d |
|- ( N e. NN -> ( sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) + 0 ) = ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + 0 ) ) |
92 |
|
fzfid |
|- ( N e. NN -> ( 1 ... N ) e. Fin ) |
93 |
26
|
a1i |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> -u 1 e. CC ) |
94 |
|
elfznn |
|- ( k e. ( 1 ... N ) -> k e. NN ) |
95 |
|
nnm1nn0 |
|- ( k e. NN -> ( k - 1 ) e. NN0 ) |
96 |
94 95
|
syl |
|- ( k e. ( 1 ... N ) -> ( k - 1 ) e. NN0 ) |
97 |
96
|
adantl |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( k - 1 ) e. NN0 ) |
98 |
93 97
|
expcld |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( -u 1 ^ ( k - 1 ) ) e. CC ) |
99 |
|
elfzelz |
|- ( k e. ( 1 ... N ) -> k e. ZZ ) |
100 |
|
elfzel1 |
|- ( k e. ( 1 ... N ) -> 1 e. ZZ ) |
101 |
99 100
|
zsubcld |
|- ( k e. ( 1 ... N ) -> ( k - 1 ) e. ZZ ) |
102 |
13 101 19
|
syl2an |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. NN0 ) |
103 |
102
|
nn0cnd |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( N - 1 ) _C ( k - 1 ) ) e. CC ) |
104 |
98 103
|
mulcld |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) e. CC ) |
105 |
92 104
|
fsumcl |
|- ( N e. NN -> sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) e. CC ) |
106 |
105
|
addid1d |
|- ( N e. NN -> ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + 0 ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
107 |
91 106
|
eqtrd |
|- ( N e. NN -> ( sum_ j e. ( 1 ... N ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) + 0 ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
108 |
66 84 107
|
3eqtrd |
|- ( N e. NN -> sum_ j e. ( 1 ... ( N + 1 ) ) ( ( -u 1 ^ ( j - 1 ) ) x. ( ( N - 1 ) _C ( j - 1 ) ) ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
109 |
42 46 108
|
3eqtrd |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) = sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
110 |
|
elnn0uz |
|- ( N e. NN0 <-> N e. ( ZZ>= ` 0 ) ) |
111 |
70 110
|
sylib |
|- ( N e. NN -> N e. ( ZZ>= ` 0 ) ) |
112 |
|
oveq2 |
|- ( k = 0 -> ( -u 1 ^ k ) = ( -u 1 ^ 0 ) ) |
113 |
|
oveq1 |
|- ( k = 0 -> ( k - 1 ) = ( 0 - 1 ) ) |
114 |
113
|
oveq2d |
|- ( k = 0 -> ( ( N - 1 ) _C ( k - 1 ) ) = ( ( N - 1 ) _C ( 0 - 1 ) ) ) |
115 |
112 114
|
oveq12d |
|- ( k = 0 -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( ( -u 1 ^ 0 ) x. ( ( N - 1 ) _C ( 0 - 1 ) ) ) ) |
116 |
111 34 115
|
fsum1p |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( ( ( -u 1 ^ 0 ) x. ( ( N - 1 ) _C ( 0 - 1 ) ) ) + sum_ k e. ( ( 0 + 1 ) ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
117 |
27
|
exp0d |
|- ( N e. NN -> ( -u 1 ^ 0 ) = 1 ) |
118 |
|
0z |
|- 0 e. ZZ |
119 |
|
zsubcl |
|- ( ( 0 e. ZZ /\ 1 e. ZZ ) -> ( 0 - 1 ) e. ZZ ) |
120 |
118 30 119
|
mp2an |
|- ( 0 - 1 ) e. ZZ |
121 |
120
|
a1i |
|- ( N e. NN -> ( 0 - 1 ) e. ZZ ) |
122 |
|
0re |
|- 0 e. RR |
123 |
|
ltm1 |
|- ( 0 e. RR -> ( 0 - 1 ) < 0 ) |
124 |
122 123
|
mp1i |
|- ( N e. NN -> ( 0 - 1 ) < 0 ) |
125 |
124
|
orcd |
|- ( N e. NN -> ( ( 0 - 1 ) < 0 \/ ( N - 1 ) < ( 0 - 1 ) ) ) |
126 |
|
bcval4 |
|- ( ( ( N - 1 ) e. NN0 /\ ( 0 - 1 ) e. ZZ /\ ( ( 0 - 1 ) < 0 \/ ( N - 1 ) < ( 0 - 1 ) ) ) -> ( ( N - 1 ) _C ( 0 - 1 ) ) = 0 ) |
127 |
13 121 125 126
|
syl3anc |
|- ( N e. NN -> ( ( N - 1 ) _C ( 0 - 1 ) ) = 0 ) |
128 |
117 127
|
oveq12d |
|- ( N e. NN -> ( ( -u 1 ^ 0 ) x. ( ( N - 1 ) _C ( 0 - 1 ) ) ) = ( 1 x. 0 ) ) |
129 |
6
|
a1i |
|- ( N e. NN -> 1 e. CC ) |
130 |
129
|
mul01d |
|- ( N e. NN -> ( 1 x. 0 ) = 0 ) |
131 |
128 130
|
eqtrd |
|- ( N e. NN -> ( ( -u 1 ^ 0 ) x. ( ( N - 1 ) _C ( 0 - 1 ) ) ) = 0 ) |
132 |
43
|
a1i |
|- ( N e. NN -> ( 0 + 1 ) = 1 ) |
133 |
132
|
oveq1d |
|- ( N e. NN -> ( ( 0 + 1 ) ... N ) = ( 1 ... N ) ) |
134 |
99
|
zcnd |
|- ( k e. ( 1 ... N ) -> k e. CC ) |
135 |
|
npcan1 |
|- ( k e. CC -> ( ( k - 1 ) + 1 ) = k ) |
136 |
135
|
eqcomd |
|- ( k e. CC -> k = ( ( k - 1 ) + 1 ) ) |
137 |
134 136
|
syl |
|- ( k e. ( 1 ... N ) -> k = ( ( k - 1 ) + 1 ) ) |
138 |
137
|
adantl |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> k = ( ( k - 1 ) + 1 ) ) |
139 |
138
|
oveq2d |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( -u 1 ^ k ) = ( -u 1 ^ ( ( k - 1 ) + 1 ) ) ) |
140 |
|
expp1 |
|- ( ( -u 1 e. CC /\ ( k - 1 ) e. NN0 ) -> ( -u 1 ^ ( ( k - 1 ) + 1 ) ) = ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) ) |
141 |
27 96 140
|
syl2an |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( -u 1 ^ ( ( k - 1 ) + 1 ) ) = ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) ) |
142 |
139 141
|
eqtrd |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( -u 1 ^ k ) = ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) ) |
143 |
142
|
oveq1d |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
144 |
98 93
|
mulcomd |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) = ( -u 1 x. ( -u 1 ^ ( k - 1 ) ) ) ) |
145 |
144
|
oveq1d |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( ( -u 1 ^ ( k - 1 ) ) x. -u 1 ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( ( -u 1 x. ( -u 1 ^ ( k - 1 ) ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
146 |
93 98 103
|
mulassd |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( -u 1 x. ( -u 1 ^ ( k - 1 ) ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( -u 1 x. ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
147 |
143 145 146
|
3eqtrd |
|- ( ( N e. NN /\ k e. ( 1 ... N ) ) -> ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( -u 1 x. ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
148 |
133 147
|
sumeq12rdv |
|- ( N e. NN -> sum_ k e. ( ( 0 + 1 ) ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = sum_ k e. ( 1 ... N ) ( -u 1 x. ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
149 |
92 27 104
|
fsummulc2 |
|- ( N e. NN -> ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = sum_ k e. ( 1 ... N ) ( -u 1 x. ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
150 |
148 149
|
eqtr4d |
|- ( N e. NN -> sum_ k e. ( ( 0 + 1 ) ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
151 |
131 150
|
oveq12d |
|- ( N e. NN -> ( ( ( -u 1 ^ 0 ) x. ( ( N - 1 ) _C ( 0 - 1 ) ) ) + sum_ k e. ( ( 0 + 1 ) ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = ( 0 + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) ) |
152 |
27 105
|
mulcld |
|- ( N e. NN -> ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) e. CC ) |
153 |
152
|
addid2d |
|- ( N e. NN -> ( 0 + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) = ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
154 |
116 151 153
|
3eqtrd |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) = ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
155 |
109 154
|
oveq12d |
|- ( N e. NN -> ( sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) ) |
156 |
35 155
|
eqtrd |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C k ) ) + ( ( -u 1 ^ k ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) ) |
157 |
105
|
mulm1d |
|- ( N e. NN -> ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = -u sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) |
158 |
157
|
oveq2d |
|- ( N e. NN -> ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) = ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + -u sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) |
159 |
105
|
negidd |
|- ( N e. NN -> ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + -u sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) = 0 ) |
160 |
158 159
|
eqtrd |
|- ( N e. NN -> ( sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) + ( -u 1 x. sum_ k e. ( 1 ... N ) ( ( -u 1 ^ ( k - 1 ) ) x. ( ( N - 1 ) _C ( k - 1 ) ) ) ) ) = 0 ) |
161 |
24 156 160
|
3eqtrd |
|- ( N e. NN -> sum_ k e. ( 0 ... N ) ( ( -u 1 ^ k ) x. ( N _C k ) ) = 0 ) |