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

Definition df-rlim 11928
Description: Define the limit relation for partial functions on the reals. See rlim 11934 for its relational expression. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
df-rlim  |-  ~~> r  =  { <. f ,  x >.  |  ( ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y ) ) }
Distinct variable group:    x, w, y, z, f

Detailed syntax breakdown of Definition df-rlim
StepHypRef Expression
1 crli 11924 . 2  class  ~~> r
2 vf . . . . . . 7  set  f
32cv 1618 . . . . . 6  class  f
4 cc 8703 . . . . . . 7  class  CC
5 cr 8704 . . . . . . 7  class  RR
6 cpm 6741 . . . . . . 7  class  ^pm
74, 5, 6co 5792 . . . . . 6  class  ( CC 
^pm  RR )
83, 7wcel 1621 . . . . 5  wff  f  e.  ( CC  ^pm  RR )
9 vx . . . . . . 7  set  x
109cv 1618 . . . . . 6  class  x
1110, 4wcel 1621 . . . . 5  wff  x  e.  CC
128, 11wa 360 . . . 4  wff  ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )
13 vz . . . . . . . . . 10  set  z
1413cv 1618 . . . . . . . . 9  class  z
15 vw . . . . . . . . . 10  set  w
1615cv 1618 . . . . . . . . 9  class  w
17 cle 8836 . . . . . . . . 9  class  <_
1814, 16, 17wbr 3997 . . . . . . . 8  wff  z  <_  w
1916, 3cfv 4673 . . . . . . . . . . 11  class  ( f `
 w )
20 cmin 9005 . . . . . . . . . . 11  class  -
2119, 10, 20co 5792 . . . . . . . . . 10  class  ( ( f `  w )  -  x )
22 cabs 11684 . . . . . . . . . 10  class  abs
2321, 22cfv 4673 . . . . . . . . 9  class  ( abs `  ( ( f `  w )  -  x
) )
24 vy . . . . . . . . . 10  set  y
2524cv 1618 . . . . . . . . 9  class  y
26 clt 8835 . . . . . . . . 9  class  <
2723, 25, 26wbr 3997 . . . . . . . 8  wff  ( abs `  ( ( f `  w )  -  x
) )  <  y
2818, 27wi 6 . . . . . . 7  wff  ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y )
293cdm 4661 . . . . . . 7  class  dom  f
3028, 15, 29wral 2518 . . . . . 6  wff  A. w  e.  dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y )
3130, 13, 5wrex 2519 . . . . 5  wff  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y )
32 crp 10321 . . . . 5  class  RR+
3331, 24, 32wral 2518 . . . 4  wff  A. y  e.  RR+  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y )
3412, 33wa 360 . . 3  wff  ( ( f  e.  ( CC 
^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y ) )
3534, 2, 9copab 4050 . 2  class  { <. f ,  x >.  |  ( ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e.  dom  f
( z  <_  w  ->  ( abs `  (
( f `  w
)  -  x ) )  <  y ) ) }
361, 35wceq 1619 1  wff  ~~> r  =  { <. f ,  x >.  |  ( ( f  e.  ( CC  ^pm  RR )  /\  x  e.  CC )  /\  A. y  e.  RR+  E. z  e.  RR  A. w  e. 
dom  f ( z  <_  w  ->  ( abs `  ( ( f `
 w )  -  x ) )  < 
y ) ) }
Colors of variables: wff set class
This definition is referenced by:  rlimrel  11932  rlim  11934  rlimpm  11939
  Copyright terms: Public domain W3C validator