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Definition df-o1 11964
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 11960 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1622 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1622 . . . . . . . . 9  class  f
63, 5cfv 5255 . . . . . . . 8  class  ( f `
 y )
7 cabs 11719 . . . . . . . 8  class  abs
86, 7cfv 5255 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1622 . . . . . . 7  class  m
11 cle 8868 . . . . . . 7  class  <_
128, 10, 11wbr 4023 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4689 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1622 . . . . . . . 8  class  x
16 cpnf 8864 . . . . . . . 8  class  +oo
17 cico 10658 . . . . . . . 8  class  [,)
1815, 16, 17co 5858 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3151 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2543 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8736 . . . . 5  class  RR
2220, 9, 21wrex 2544 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2544 . . 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 8735 . . . 4  class  CC
25 cpm 6773 . . . 4  class  ^pm
2624, 21, 25co 5858 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2547 . 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 1623 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  12000
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