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

Definition df-o1 11960
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 11956 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1624 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1624 . . . . . . . . 9  class  f
63, 5cfv 5223 . . . . . . . 8  class  ( f `
 y )
7 cabs 11715 . . . . . . . 8  class  abs
86, 7cfv 5223 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1624 . . . . . . 7  class  m
11 cle 8865 . . . . . . 7  class  <_
128, 10, 11wbr 4026 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4690 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1624 . . . . . . . 8  class  x
16 cpnf 8861 . . . . . . . 8  class  +oo
17 cico 10654 . . . . . . . 8  class  [,)
1815, 16, 17co 5821 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3154 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2546 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8733 . . . . 5  class  RR
2220, 9, 21wrex 2547 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2547 . . 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 8732 . . . 4  class  CC
25 cpm 6770 . . . 4  class  ^pm
2624, 21, 25co 5821 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2550 . 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 1625 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  11996
  Copyright terms: Public domain W3C validator