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Definition df-itg 18974
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 18972 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 18972 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 18968 . 2  class  S. A B  _d x
5 cc0 8733 . . . 4  class  0
6 c3 9792 . . . 4  class  3
7 cfz 10777 . . . 4  class  ...
85, 6, 7co 5820 . . 3  class  ( 0 ... 3 )
9 ci 8735 . . . . 5  class  _i
10 vk . . . . . 6  set  k
1110cv 1623 . . . . 5  class  k
12 cexp 11099 . . . . 5  class  ^
139, 11, 12co 5820 . . . 4  class  ( _i
^ k )
14 cr 8732 . . . . . 6  class  RR
15 vy . . . . . . 7  set  y
16 cdiv 9419 . . . . . . . . 9  class  /
173, 13, 16co 5820 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 11577 . . . . . . . 8  class  Re
1917, 18cfv 5222 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1623 . . . . . . . . . 10  class  x
2120, 2wcel 1685 . . . . . . . . 9  wff  x  e.  A
2215cv 1623 . . . . . . . . . 10  class  y
23 cle 8864 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4025 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 360 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3567 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3083 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4079 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 18966 . . . . 5  class  S.2
3028, 29cfv 5222 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 8738 . . . 4  class  x.
3213, 30, 31co 5820 . . 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 12153 . 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 1624 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  19119  itgex  19120  nfitg1  19123
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