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Metamath Proof Explorer

Theorem itg0

Description: The integral of anything on the empty set is zero. (Contributed by Mario Carneiro, 13-Aug-2014)

Ref Expression
Assertion itg0 
|- S. (/) A _d x = 0

Proof

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 eqtrid 
 |-  ( 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