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Definition df-o1 11929
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 11925 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1618 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1618 . . . . . . . . 9  class  f
63, 5cfv 4673 . . . . . . . 8  class  ( f `
 y )
7 cabs 11684 . . . . . . . 8  class  abs
86, 7cfv 4673 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1618 . . . . . . 7  class  m
11 cle 8836 . . . . . . 7  class  <_
128, 10, 11wbr 3997 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4661 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1618 . . . . . . . 8  class  x
16 cpnf 8832 . . . . . . . 8  class  +oo
17 cico 10624 . . . . . . . 8  class  [,)
1815, 16, 17co 5792 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3126 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2518 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8704 . . . . 5  class  RR
2220, 9, 21wrex 2519 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2519 . . 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 8703 . . . 4  class  CC
25 cpm 6741 . . . 4  class  ^pm
2624, 21, 25co 5792 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2522 . 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 1619 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  11965
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