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Definition df-itg 21118
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 21116 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 21116 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 21113 . 2  class  S. A B  _d x
5 cc0 9297 . . . 4  class  0
6 c3 10387 . . . 4  class  3
7 cfz 11452 . . . 4  class  ...
85, 6, 7co 6106 . . 3  class  ( 0 ... 3 )
9 ci 9299 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1368 . . . . 5  class  k
12 cexp 11880 . . . . 5  class  ^
139, 11, 12co 6106 . . . 4  class  ( _i
^ k )
14 cr 9296 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 10008 . . . . . . . . 9  class  /
173, 13, 16co 6106 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 12601 . . . . . . . 8  class  Re
1917, 18cfv 5433 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1368 . . . . . . . . . 10  class  x
2120, 2wcel 1756 . . . . . . . . 9  wff  x  e.  A
2215cv 1368 . . . . . . . . . 10  class  y
23 cle 9434 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4307 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 369 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3806 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3303 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4365 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 21111 . . . . 5  class  S.2
3028, 29cfv 5433 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9302 . . . 4  class  x.
3213, 30, 31co 6106 . . 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 13178 . 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 1369 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  21262  itgex  21263  nfitg1  21266
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