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Definition df-o1 12964
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 12960 . 2  class  O(1)
2 vy . . . . . . . . . 10  setvar  y
32cv 1363 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  setvar  f
54cv 1363 . . . . . . . . 9  class  f
63, 5cfv 5415 . . . . . . . 8  class  ( f `
 y )
7 cabs 12719 . . . . . . . 8  class  abs
86, 7cfv 5415 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  setvar  m
109cv 1363 . . . . . . 7  class  m
11 cle 9415 . . . . . . 7  class  <_
128, 10, 11wbr 4289 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4836 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  setvar  x
1514cv 1363 . . . . . . . 8  class  x
16 cpnf 9411 . . . . . . . 8  class +oo
17 cico 11298 . . . . . . . 8  class  [,)
1815, 16, 17co 6090 . . . . . . 7  class  ( x [,) +oo )
1913, 18cin 3324 . . . . . 6  class  ( dom  f  i^i  ( x [,) +oo ) )
2012, 2, 19wral 2713 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 9277 . . . . 5  class  RR
2220, 9, 21wrex 2714 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2714 . . 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 9276 . . . 4  class  CC
25 cpm 7211 . . . 4  class  ^pm
2624, 21, 25co 6090 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2717 . 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 1364 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 setvar class
This definition is referenced by:  elo1  13000
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