MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-o1 Unicode version

Definition df-o1 12211
Description: Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of  O ( 1 ) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
df-o1  |-  O ( 1 )  =  {
f  e.  ( CC 
^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m }
Distinct variable group:    x, y, f, m

Detailed syntax breakdown of Definition df-o1
StepHypRef Expression
1 co1 12207 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1648 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1648 . . . . . . . . 9  class  f
63, 5cfv 5394 . . . . . . . 8  class  ( f `
 y )
7 cabs 11966 . . . . . . . 8  class  abs
86, 7cfv 5394 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1648 . . . . . . 7  class  m
11 cle 9054 . . . . . . 7  class  <_
128, 10, 11wbr 4153 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4818 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1648 . . . . . . . 8  class  x
16 cpnf 9050 . . . . . . . 8  class  +oo
17 cico 10850 . . . . . . . 8  class  [,)
1815, 16, 17co 6020 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3262 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2649 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8922 . . . . 5  class  RR
2220, 9, 21wrex 2650 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2650 . . 3  wff  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
24 cc 8921 . . . 4  class  CC
25 cpm 6955 . . . 4  class  ^pm
2624, 21, 25co 6020 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2653 . 2  class  { f  e.  ( CC  ^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m }
281, 27wceq 1649 1  wff  O ( 1 )  =  {
f  e.  ( CC 
^pm  RR )  |  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m }
Colors of variables: wff set class
This definition is referenced by:  elo1  12247
  Copyright terms: Public domain W3C validator