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Definition df-itg 21062
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 21060 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 21060 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 21057 . 2  class  S. A B  _d x
5 cc0 9278 . . . 4  class  0
6 c3 10368 . . . 4  class  3
7 cfz 11433 . . . 4  class  ...
85, 6, 7co 6090 . . 3  class  ( 0 ... 3 )
9 ci 9280 . . . . 5  class  _i
10 vk . . . . . 6  setvar  k
1110cv 1363 . . . . 5  class  k
12 cexp 11861 . . . . 5  class  ^
139, 11, 12co 6090 . . . 4  class  ( _i
^ k )
14 cr 9277 . . . . . 6  class  RR
15 vy . . . . . . 7  setvar  y
16 cdiv 9989 . . . . . . . . 9  class  /
173, 13, 16co 6090 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 12582 . . . . . . . 8  class  Re
1917, 18cfv 5415 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1363 . . . . . . . . . 10  class  x
2120, 2wcel 1761 . . . . . . . . 9  wff  x  e.  A
2215cv 1363 . . . . . . . . . 10  class  y
23 cle 9415 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4289 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 369 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3788 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3285 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4347 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 21055 . . . . 5  class  S.2
3028, 29cfv 5415 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 9283 . . . 4  class  x.
3213, 30, 31co 6090 . . 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 13159 . 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 1364 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  21206  itgex  21207  nfitg1  21210
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