Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Re ` ( A / ( _i ^ k ) ) ) = ( Re ` ( A / ( _i ^ k ) ) ) |
2 |
1
|
dfitg |
|- S. (/) A _d x = sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) ) |
3 |
|
ifan |
|- if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) = if ( x e. (/) , if ( 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) , 0 ) |
4 |
|
noel |
|- -. x e. (/) |
5 |
4
|
iffalsei |
|- if ( x e. (/) , if ( 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) , 0 ) = 0 |
6 |
3 5
|
eqtri |
|- if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) = 0 |
7 |
6
|
mpteq2i |
|- ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) = ( x e. RR |-> 0 ) |
8 |
|
fconstmpt |
|- ( RR X. { 0 } ) = ( x e. RR |-> 0 ) |
9 |
7 8
|
eqtr4i |
|- ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) = ( RR X. { 0 } ) |
10 |
9
|
fveq2i |
|- ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) = ( S.2 ` ( RR X. { 0 } ) ) |
11 |
|
itg20 |
|- ( S.2 ` ( RR X. { 0 } ) ) = 0 |
12 |
10 11
|
eqtri |
|- ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) = 0 |
13 |
12
|
oveq2i |
|- ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) ) = ( ( _i ^ k ) x. 0 ) |
14 |
|
ax-icn |
|- _i e. CC |
15 |
|
elfznn0 |
|- ( k e. ( 0 ... 3 ) -> k e. NN0 ) |
16 |
|
expcl |
|- ( ( _i e. CC /\ k e. NN0 ) -> ( _i ^ k ) e. CC ) |
17 |
14 15 16
|
sylancr |
|- ( k e. ( 0 ... 3 ) -> ( _i ^ k ) e. CC ) |
18 |
17
|
mul01d |
|- ( k e. ( 0 ... 3 ) -> ( ( _i ^ k ) x. 0 ) = 0 ) |
19 |
13 18
|
syl5eq |
|- ( k e. ( 0 ... 3 ) -> ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) ) = 0 ) |
20 |
19
|
sumeq2i |
|- sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) ) = sum_ k e. ( 0 ... 3 ) 0 |
21 |
|
fzfi |
|- ( 0 ... 3 ) e. Fin |
22 |
21
|
olci |
|- ( ( 0 ... 3 ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... 3 ) e. Fin ) |
23 |
|
sumz |
|- ( ( ( 0 ... 3 ) C_ ( ZZ>= ` 0 ) \/ ( 0 ... 3 ) e. Fin ) -> sum_ k e. ( 0 ... 3 ) 0 = 0 ) |
24 |
22 23
|
ax-mp |
|- sum_ k e. ( 0 ... 3 ) 0 = 0 |
25 |
20 24
|
eqtri |
|- sum_ k e. ( 0 ... 3 ) ( ( _i ^ k ) x. ( S.2 ` ( x e. RR |-> if ( ( x e. (/) /\ 0 <_ ( Re ` ( A / ( _i ^ k ) ) ) ) , ( Re ` ( A / ( _i ^ k ) ) ) , 0 ) ) ) ) = 0 |
26 |
2 25
|
eqtri |
|- S. (/) A _d x = 0 |