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Definition df-itg 19469
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 19467 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 19467 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 19463 . 2  class  S. A B  _d x
5 cc0 8946 . . . 4  class  0
6 c3 10006 . . . 4  class  3
7 cfz 10999 . . . 4  class  ...
85, 6, 7co 6040 . . 3  class  ( 0 ... 3 )
9 ci 8948 . . . . 5  class  _i
10 vk . . . . . 6  set  k
1110cv 1648 . . . . 5  class  k
12 cexp 11337 . . . . 5  class  ^
139, 11, 12co 6040 . . . 4  class  ( _i
^ k )
14 cr 8945 . . . . . 6  class  RR
15 vy . . . . . . 7  set  y
16 cdiv 9633 . . . . . . . . 9  class  /
173, 13, 16co 6040 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 11857 . . . . . . . 8  class  Re
1917, 18cfv 5413 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1648 . . . . . . . . . 10  class  x
2120, 2wcel 1721 . . . . . . . . 9  wff  x  e.  A
2215cv 1648 . . . . . . . . . 10  class  y
23 cle 9077 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4172 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 359 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3699 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3211 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4226 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 19461 . . . . 5  class  S.2
3028, 29cfv 5413 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 8951 . . . 4  class  x.
3213, 30, 31co 6040 . . 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 12434 . 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 1649 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  19614  itgex  19615  nfitg1  19618
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