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

Definition df-vol 20790
Description: Define the Lebesgue measure, which is just the outer measure with a peculiar domain of definition. The property of being Lebesgue-measurable can be expressed as  A  e.  dom  vol. (Contributed by Mario Carneiro, 17-Mar-2014.)
Assertion
Ref Expression
df-vol  |-  vol  =  ( vol*  |`  { x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-vol
StepHypRef Expression
1 cvol 20788 . 2  class  vol
2 covol 20787 . . 3  class  vol*
3 vy . . . . . . . 8  setvar  y
43cv 1361 . . . . . . 7  class  y
54, 2cfv 5406 . . . . . 6  class  ( vol* `  y )
6 vx . . . . . . . . . 10  setvar  x
76cv 1361 . . . . . . . . 9  class  x
84, 7cin 3315 . . . . . . . 8  class  ( y  i^i  x )
98, 2cfv 5406 . . . . . . 7  class  ( vol* `  ( y  i^i  x ) )
104, 7cdif 3313 . . . . . . . 8  class  ( y 
\  x )
1110, 2cfv 5406 . . . . . . 7  class  ( vol* `  ( y  \  x ) )
12 caddc 9272 . . . . . . 7  class  +
139, 11, 12co 6080 . . . . . 6  class  ( ( vol* `  (
y  i^i  x )
)  +  ( vol* `  ( y  \  x ) ) )
145, 13wceq 1362 . . . . 5  wff  ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x ) )  +  ( vol* `  ( y  \  x
) ) )
152ccnv 4826 . . . . . 6  class  `' vol*
16 cr 9268 . . . . . 6  class  RR
1715, 16cima 4830 . . . . 5  class  ( `' vol* " RR )
1814, 3, 17wral 2705 . . . 4  wff  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x ) )  +  ( vol* `  ( y  \  x
) ) )
1918, 6cab 2419 . . 3  class  { x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) }
202, 19cres 4829 . 2  class  ( vol*  |`  { x  | 
A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
211, 20wceq 1362 1  wff  vol  =  ( vol*  |`  { x  |  A. y  e.  ( `' vol* " RR ) ( vol* `  y )  =  ( ( vol* `  ( y  i^i  x
) )  +  ( vol* `  (
y  \  x )
) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  ismbl  20850  volres  20852
  Copyright terms: Public domain W3C validator