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Definition df-itg 19499
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 19497 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 19497 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  set  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3citg 19493 . 2  class  S. A B  _d x
5 cc0 8974 . . . 4  class  0
6 c3 10034 . . . 4  class  3
7 cfz 11027 . . . 4  class  ...
85, 6, 7co 6067 . . 3  class  ( 0 ... 3 )
9 ci 8976 . . . . 5  class  _i
10 vk . . . . . 6  set  k
1110cv 1651 . . . . 5  class  k
12 cexp 11365 . . . . 5  class  ^
139, 11, 12co 6067 . . . 4  class  ( _i
^ k )
14 cr 8973 . . . . . 6  class  RR
15 vy . . . . . . 7  set  y
16 cdiv 9661 . . . . . . . . 9  class  /
173, 13, 16co 6067 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 11885 . . . . . . . 8  class  Re
1917, 18cfv 5440 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1651 . . . . . . . . . 10  class  x
2120, 2wcel 1725 . . . . . . . . 9  wff  x  e.  A
2215cv 1651 . . . . . . . . . 10  class  y
23 cle 9105 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4199 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 359 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3726 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3238 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4253 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 19491 . . . . 5  class  S.2
3028, 29cfv 5440 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 8979 . . . 4  class  x.
3213, 30, 31co 6067 . . 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 12462 . 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 1652 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 set class
This definition is referenced by:  dfitg  19644  itgex  19645  nfitg1  19648
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