Step |
Hyp |
Ref |
Expression |
1 |
|
1nn0 |
|- 1 e. NN0 |
2 |
|
bpolyval |
|- ( ( 1 e. NN0 /\ X e. CC ) -> ( 1 BernPoly X ) = ( ( X ^ 1 ) - sum_ k e. ( 0 ... ( 1 - 1 ) ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) ) ) |
3 |
1 2
|
mpan |
|- ( X e. CC -> ( 1 BernPoly X ) = ( ( X ^ 1 ) - sum_ k e. ( 0 ... ( 1 - 1 ) ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) ) ) |
4 |
|
exp1 |
|- ( X e. CC -> ( X ^ 1 ) = X ) |
5 |
|
1m1e0 |
|- ( 1 - 1 ) = 0 |
6 |
5
|
oveq2i |
|- ( 0 ... ( 1 - 1 ) ) = ( 0 ... 0 ) |
7 |
6
|
sumeq1i |
|- sum_ k e. ( 0 ... ( 1 - 1 ) ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = sum_ k e. ( 0 ... 0 ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) |
8 |
|
0z |
|- 0 e. ZZ |
9 |
|
bpoly0 |
|- ( X e. CC -> ( 0 BernPoly X ) = 1 ) |
10 |
9
|
oveq1d |
|- ( X e. CC -> ( ( 0 BernPoly X ) / 2 ) = ( 1 / 2 ) ) |
11 |
10
|
oveq2d |
|- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) = ( 1 x. ( 1 / 2 ) ) ) |
12 |
|
halfcn |
|- ( 1 / 2 ) e. CC |
13 |
12
|
mulid2i |
|- ( 1 x. ( 1 / 2 ) ) = ( 1 / 2 ) |
14 |
11 13
|
eqtrdi |
|- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) = ( 1 / 2 ) ) |
15 |
14 12
|
eqeltrdi |
|- ( X e. CC -> ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) e. CC ) |
16 |
|
oveq2 |
|- ( k = 0 -> ( 1 _C k ) = ( 1 _C 0 ) ) |
17 |
|
bcn0 |
|- ( 1 e. NN0 -> ( 1 _C 0 ) = 1 ) |
18 |
1 17
|
ax-mp |
|- ( 1 _C 0 ) = 1 |
19 |
16 18
|
eqtrdi |
|- ( k = 0 -> ( 1 _C k ) = 1 ) |
20 |
|
oveq1 |
|- ( k = 0 -> ( k BernPoly X ) = ( 0 BernPoly X ) ) |
21 |
|
oveq2 |
|- ( k = 0 -> ( 1 - k ) = ( 1 - 0 ) ) |
22 |
|
1m0e1 |
|- ( 1 - 0 ) = 1 |
23 |
21 22
|
eqtrdi |
|- ( k = 0 -> ( 1 - k ) = 1 ) |
24 |
23
|
oveq1d |
|- ( k = 0 -> ( ( 1 - k ) + 1 ) = ( 1 + 1 ) ) |
25 |
|
df-2 |
|- 2 = ( 1 + 1 ) |
26 |
24 25
|
eqtr4di |
|- ( k = 0 -> ( ( 1 - k ) + 1 ) = 2 ) |
27 |
20 26
|
oveq12d |
|- ( k = 0 -> ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) = ( ( 0 BernPoly X ) / 2 ) ) |
28 |
19 27
|
oveq12d |
|- ( k = 0 -> ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) ) |
29 |
28
|
fsum1 |
|- ( ( 0 e. ZZ /\ ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) e. CC ) -> sum_ k e. ( 0 ... 0 ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) ) |
30 |
8 15 29
|
sylancr |
|- ( X e. CC -> sum_ k e. ( 0 ... 0 ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = ( 1 x. ( ( 0 BernPoly X ) / 2 ) ) ) |
31 |
30 14
|
eqtrd |
|- ( X e. CC -> sum_ k e. ( 0 ... 0 ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = ( 1 / 2 ) ) |
32 |
7 31
|
eqtrid |
|- ( X e. CC -> sum_ k e. ( 0 ... ( 1 - 1 ) ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) = ( 1 / 2 ) ) |
33 |
4 32
|
oveq12d |
|- ( X e. CC -> ( ( X ^ 1 ) - sum_ k e. ( 0 ... ( 1 - 1 ) ) ( ( 1 _C k ) x. ( ( k BernPoly X ) / ( ( 1 - k ) + 1 ) ) ) ) = ( X - ( 1 / 2 ) ) ) |
34 |
3 33
|
eqtrd |
|- ( X e. CC -> ( 1 BernPoly X ) = ( X - ( 1 / 2 ) ) ) |