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Definition df-o1 12289
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 12285 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1652 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1652 . . . . . . . . 9  class  f
63, 5cfv 5457 . . . . . . . 8  class  ( f `
 y )
7 cabs 12044 . . . . . . . 8  class  abs
86, 7cfv 5457 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1652 . . . . . . 7  class  m
11 cle 9126 . . . . . . 7  class  <_
128, 10, 11wbr 4215 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4881 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1652 . . . . . . . 8  class  x
16 cpnf 9122 . . . . . . . 8  class  +oo
17 cico 10923 . . . . . . . 8  class  [,)
1815, 16, 17co 6084 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3321 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2707 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8994 . . . . 5  class  RR
2220, 9, 21wrex 2708 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2708 . . 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 8993 . . . 4  class  CC
25 cpm 7022 . . . 4  class  ^pm
2624, 21, 25co 6084 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2711 . 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 1653 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  12325
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