MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-itg Structured version   Unicode version

Definition df-itg 21767
Description: Define the full Lebesgue integral, for complex-valued functions to  RR. The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of  x ^ 2 from  0 to  1 is  S. ( 0 [,] 1 ) ( x ^ 2 )  _d x  =  ( 1  /  3 ). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 21765 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 21765 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)
Assertion
Ref Expression
df-itg  |-  S. A B  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )
Distinct variable groups:    y, k, A    B, k, y    x, k, y
Allowed substitution hints:    A( x)    B( x)

Detailed syntax breakdown of Definition df-itg
StepHypRef Expression
1 vx . . 3  setvar  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3citg 21762 . 2  class  S. A B  _d x
5 cc0 9488 . . . 4  class  0
6 c3 10582 . . . 4  class  3
7 cfz 11668 . . . 4  class  ...
85, 6, 7co 6282 . . 3  class  ( 0 ... 3 )
9 ci 9490 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1378 . . . . 5  class  k
12 cexp 12130 . . . . 5  class  ^
139, 11, 12co 6282 . . . 4  class  ( _i
^ k )
14 cr 9487 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 10202 . . . . . . . . 9  class  /
173, 13, 16co 6282 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 12889 . . . . . . . 8  class  Re
1917, 18cfv 5586 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1378 . . . . . . . . . 10  class  x
2120, 2wcel 1767 . . . . . . . . 9  wff  x  e.  A
2215cv 1378 . . . . . . . . . 10  class  y
23 cle 9625 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4447 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 369 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3939 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3435 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4505 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 21760 . . . . 5  class  S.2
3028, 29cfv 5586 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9493 . . . 4  class  x.
3213, 30, 31co 6282 . . 3  class  ( ( _i ^ k )  x.  ( S.2 `  (
x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )
338, 32, 10csu 13467 . 2  class  sum_ k  e.  ( 0 ... 3
) ( ( _i
^ k )  x.  ( S.2 `  (
x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )
344, 33wceq 1379 1  wff  S. A B  _d x  =  sum_ k  e.  ( 0 ... 3 ) ( ( _i ^ k
)  x.  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  dfitg  21911  itgex  21912  nfitg1  21915
  Copyright terms: Public domain W3C validator