Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Re ` ( 0 / ( _i ^ k ) ) ) = ( Re ` ( 0 / ( _i ^ k ) ) ) |
2 |
1
|
dfitg |
|- S. A 0 _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) ) |
3 |
|
ax-icn |
|- _i e. CC |
4 |
|
elfznn0 |
|- ( k e. ( 0 ... 3 ) -> k e. NN0 ) |
5 |
|
expcl |
|- ( ( _i e. CC /\ k e. NN0 ) -> ( _i ^ k ) e. CC ) |
6 |
3 4 5
|
sylancr |
|- ( k e. ( 0 ... 3 ) -> ( _i ^ k ) e. CC ) |
7 |
|
ine0 |
|- _i =/= 0 |
8 |
|
elfzelz |
|- ( k e. ( 0 ... 3 ) -> k e. ZZ ) |
9 |
|
expne0i |
|- ( ( _i e. CC /\ _i =/= 0 /\ k e. ZZ ) -> ( _i ^ k ) =/= 0 ) |
10 |
3 7 8 9
|
mp3an12i |
|- ( k e. ( 0 ... 3 ) -> ( _i ^ k ) =/= 0 ) |
11 |
6 10
|
div0d |
|- ( k e. ( 0 ... 3 ) -> ( 0 / ( _i ^ k ) ) = 0 ) |
12 |
11
|
fveq2d |
|- ( k e. ( 0 ... 3 ) -> ( Re ` ( 0 / ( _i ^ k ) ) ) = ( Re ` 0 ) ) |
13 |
|
re0 |
|- ( Re ` 0 ) = 0 |
14 |
12 13
|
eqtrdi |
|- ( k e. ( 0 ... 3 ) -> ( Re ` ( 0 / ( _i ^ k ) ) ) = 0 ) |
15 |
14
|
ifeq1d |
|- ( k e. ( 0 ... 3 ) -> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) = if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , 0 , 0 ) ) |
16 |
|
ifid |
|- if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , 0 , 0 ) = 0 |
17 |
15 16
|
eqtrdi |
|- ( k e. ( 0 ... 3 ) -> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) = 0 ) |
18 |
17
|
mpteq2dv |
|- ( k e. ( 0 ... 3 ) -> ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) = ( x e. RR |-> 0 ) ) |
19 |
|
fconstmpt |
|- ( RR X. { 0 } ) = ( x e. RR |-> 0 ) |
20 |
18 19
|
eqtr4di |
|- ( k e. ( 0 ... 3 ) -> ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) = ( RR X. { 0 } ) ) |
21 |
20
|
fveq2d |
|- ( k e. ( 0 ... 3 ) -> ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) = ( S.2 ` ( RR X. { 0 } ) ) ) |
22 |
|
itg20 |
|- ( S.2 ` ( RR X. { 0 } ) ) = 0 |
23 |
21 22
|
eqtrdi |
|- ( k e. ( 0 ... 3 ) -> ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) = 0 ) |
24 |
23
|
oveq2d |
|- ( k e. ( 0 ... 3 ) -> ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) ) = ( ( _i ^ k ) x. 0 ) ) |
25 |
6
|
mul01d |
|- ( k e. ( 0 ... 3 ) -> ( ( _i ^ k ) x. 0 ) = 0 ) |
26 |
24 25
|
eqtrd |
|- ( k e. ( 0 ... 3 ) -> ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) ) = 0 ) |
27 |
26
|
sumeq2i |
|- sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. A /\ 0 <_ ( Re ` ( 0 / ( _i ^ k ) ) ) ) , ( Re ` ( 0 / ( _i ^ k ) ) ) , 0 ) ) ) ) = sum_ k e. ( 0 ... 3 ) 0 |
28 |
|
fzfi |
|- ( 0 ... 3 ) e. Fin |
29 |
28
|
olci |
|- ( ( 0 ... 3 ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... 3 ) e. Fin ) |
30 |
|
sumz |
|- ( ( ( 0 ... 3 ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... 3 ) e. Fin ) -> sum_ k e. ( 0 ... 3 ) 0 = 0 ) |
31 |
29 30
|
ax-mp |
|- sum_ k e. ( 0 ... 3 ) 0 = 0 |
32 |
2 27 31
|
3eqtri |
|- S. A 0 _d x = 0 |