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Definition df-itg 22140
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 22138 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 22138 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 22135 . 2  class  S. A B  _d x
5 cc0 9425 . . . 4  class  0
6 c3 10525 . . . 4  class  3
7 cfz 11615 . . . 4  class  ...
85, 6, 7co 6218 . . 3  class  ( 0 ... 3 )
9 ci 9427 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1398 . . . . 5  class  k
12 cexp 12092 . . . . 5  class  ^
139, 11, 12co 6218 . . . 4  class  ( _i
^ k )
14 cr 9424 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 10145 . . . . . . . . 9  class  /
173, 13, 16co 6218 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 12955 . . . . . . . 8  class  Re
1917, 18cfv 5513 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1398 . . . . . . . . . 10  class  x
2120, 2wcel 1836 . . . . . . . . 9  wff  x  e.  A
2215cv 1398 . . . . . . . . . 10  class  y
23 cle 9562 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4384 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 367 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3874 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3365 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4442 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 22133 . . . . 5  class  S.2
3028, 29cfv 5513 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9430 . . . 4  class  x.
3213, 30, 31co 6218 . . 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 13533 . 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 1399 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  22284  itgex  22285  nfitg1  22288
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