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Definition df-o1 11980
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 11976 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1631 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1631 . . . . . . . . 9  class  f
63, 5cfv 5271 . . . . . . . 8  class  ( f `
 y )
7 cabs 11735 . . . . . . . 8  class  abs
86, 7cfv 5271 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1631 . . . . . . 7  class  m
11 cle 8884 . . . . . . 7  class  <_
128, 10, 11wbr 4039 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4705 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1631 . . . . . . . 8  class  x
16 cpnf 8880 . . . . . . . 8  class  +oo
17 cico 10674 . . . . . . . 8  class  [,)
1815, 16, 17co 5874 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3164 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2556 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8752 . . . . 5  class  RR
2220, 9, 21wrex 2557 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2557 . . 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 8751 . . . 4  class  CC
25 cpm 6789 . . . 4  class  ^pm
2624, 21, 25co 5874 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2560 . 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 1632 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  12016
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