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Definition df-itg 22573
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 22571 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 22571 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 22568 . 2  class  S. A B  _d x
5 cc0 9541 . . . 4  class  0
6 c3 10662 . . . 4  class  3
7 cfz 11786 . . . 4  class  ...
85, 6, 7co 6303 . . 3  class  ( 0 ... 3 )
9 ci 9543 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1437 . . . . 5  class  k
12 cexp 12273 . . . . 5  class  ^
139, 11, 12co 6303 . . . 4  class  ( _i
^ k )
14 cr 9540 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 10271 . . . . . . . . 9  class  /
173, 13, 16co 6303 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 13154 . . . . . . . 8  class  Re
1917, 18cfv 5599 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1437 . . . . . . . . . 10  class  x
2120, 2wcel 1869 . . . . . . . . 9  wff  x  e.  A
2215cv 1437 . . . . . . . . . 10  class  y
23 cle 9678 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4421 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 371 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3910 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3396 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4480 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 22566 . . . . 5  class  S.2
3028, 29cfv 5599 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9546 . . . 4  class  x.
3213, 30, 31co 6303 . . 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 13745 . 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 1438 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  22719  itgex  22720  nfitg1  22723
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