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Definition df-itg 18979
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 18977 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 18977 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 18973 . 2  class  S. A B  _d x
5 cc0 8737 . . . 4  class  0
6 c3 9796 . . . 4  class  3
7 cfz 10782 . . . 4  class  ...
85, 6, 7co 5858 . . 3  class  ( 0 ... 3 )
9 ci 8739 . . . . 5  class  _i
10 vk . . . . . 6  set  k
1110cv 1622 . . . . 5  class  k
12 cexp 11104 . . . . 5  class  ^
139, 11, 12co 5858 . . . 4  class  ( _i
^ k )
14 cr 8736 . . . . . 6  class  RR
15 vy . . . . . . 7  set  y
16 cdiv 9423 . . . . . . . . 9  class  /
173, 13, 16co 5858 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 11582 . . . . . . . 8  class  Re
1917, 18cfv 5255 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1622 . . . . . . . . . 10  class  x
2120, 2wcel 1684 . . . . . . . . 9  wff  x  e.  A
2215cv 1622 . . . . . . . . . 10  class  y
23 cle 8868 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4023 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 358 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3565 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3081 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4077 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 18971 . . . . 5  class  S.2
3028, 29cfv 5255 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 8742 . . . 4  class  x.
3213, 30, 31co 5858 . . 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 12158 . 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 1623 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  19124  itgex  19125  nfitg1  19128
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