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Metamath Proof Explorer

Theorem itgz

Description: The integral of zero on any set is zero. (Contributed by Mario Carneiro, 29-Jun-2014) (Revised by Mario Carneiro, 23-Aug-2014)

Ref Expression
Assertion itgz 
|- S. A 0 _d x = 0

Proof

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