MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-itg Unicode version

Definition df-itg 19194
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 19192 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 19192 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 19188 . 2  class  S. A B  _d x
5 cc0 8884 . . . 4  class  0
6 c3 9943 . . . 4  class  3
7 cfz 10935 . . . 4  class  ...
85, 6, 7co 5981 . . 3  class  ( 0 ... 3 )
9 ci 8886 . . . . 5  class  _i
10 vk . . . . . 6  set  k
1110cv 1646 . . . . 5  class  k
12 cexp 11269 . . . . 5  class  ^
139, 11, 12co 5981 . . . 4  class  ( _i
^ k )
14 cr 8883 . . . . . 6  class  RR
15 vy . . . . . . 7  set  y
16 cdiv 9570 . . . . . . . . 9  class  /
173, 13, 16co 5981 . . . . . . . 8  class  ( B  /  ( _i ^
k ) )
18 cre 11789 . . . . . . . 8  class  Re
1917, 18cfv 5358 . . . . . . 7  class  ( Re
`  ( B  / 
( _i ^ k
) ) )
201cv 1646 . . . . . . . . . 10  class  x
2120, 2wcel 1715 . . . . . . . . 9  wff  x  e.  A
2215cv 1646 . . . . . . . . . 10  class  y
23 cle 9015 . . . . . . . . . 10  class  <_
245, 22, 23wbr 4125 . . . . . . . . 9  wff  0  <_  y
2521, 24wa 358 . . . . . . . 8  wff  ( x  e.  A  /\  0  <_  y )
2625, 22, 5cif 3654 . . . . . . 7  class  if ( ( x  e.  A  /\  0  <_  y ) ,  y ,  0 )
2715, 19, 26csb 3167 . . . . . 6  class  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 )
281, 14, 27cmpt 4179 . . . . 5  class  ( x  e.  RR  |->  [_ (
Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) )
29 citg2 19186 . . . . 5  class  S.2
3028, 29cfv 5358 . . . 4  class  ( S.2 `  ( x  e.  RR  |->  [_ ( Re `  ( B  /  ( _i ^
k ) ) )  /  y ]_ if ( ( x  e.  A  /\  0  <_ 
y ) ,  y ,  0 ) ) )
31 cmul 8889 . . . 4  class  x.
3213, 30, 31co 5981 . . 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 12366 . 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 1647 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  19339  itgex  19340  nfitg1  19343
  Copyright terms: Public domain W3C validator