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Definition df-o1 12284
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 12280 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1651 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1651 . . . . . . . . 9  class  f
63, 5cfv 5454 . . . . . . . 8  class  ( f `
 y )
7 cabs 12039 . . . . . . . 8  class  abs
86, 7cfv 5454 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1651 . . . . . . 7  class  m
11 cle 9121 . . . . . . 7  class  <_
128, 10, 11wbr 4212 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4878 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1651 . . . . . . . 8  class  x
16 cpnf 9117 . . . . . . . 8  class  +oo
17 cico 10918 . . . . . . . 8  class  [,)
1815, 16, 17co 6081 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3319 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2705 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8989 . . . . 5  class  RR
2220, 9, 21wrex 2706 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2706 . . 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 8988 . . . 4  class  CC
25 cpm 7019 . . . 4  class  ^pm
2624, 21, 25co 6081 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2709 . 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 1652 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  12320
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