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Definition df-itg 22581
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 22579 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 22579 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 22576 . 2  class  S. A B  _d x
5 cc0 9539 . . . 4  class  0
6 c3 10660 . . . 4  class  3
7 cfz 11784 . . . 4  class  ...
85, 6, 7co 6290 . . 3  class  ( 0 ... 3 )
9 ci 9541 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1443 . . . . 5  class  k
12 cexp 12272 . . . . 5  class  ^
139, 11, 12co 6290 . . . 4  class  ( _i
^ k )
14 cr 9538 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 10269 . . . . . . . . 9  class  /
173, 13, 16co 6290 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 13160 . . . . . . . 8  class  Re
1917, 18cfv 5582 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1443 . . . . . . . . . 10  class  x
2120, 2wcel 1887 . . . . . . . . 9  wff  x  e.  A
2215cv 1443 . . . . . . . . . 10  class  y
23 cle 9676 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4402 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 371 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3881 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3363 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4461 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 22574 . . . . 5  class  S.2
3028, 29cfv 5582 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9544 . . . 4  class  x.
3213, 30, 31co 6290 . . 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 13752 . 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 1444 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  22727  itgex  22728  nfitg1  22731
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