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Definition df-o1 13272
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 13268 . 2  class  O(1)
2 vy . . . . . . . . . 10  setvar  y
32cv 1378 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  setvar  f
54cv 1378 . . . . . . . . 9  class  f
63, 5cfv 5586 . . . . . . . 8  class  ( f `
 y )
7 cabs 13026 . . . . . . . 8  class  abs
86, 7cfv 5586 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  setvar  m
109cv 1378 . . . . . . 7  class  m
11 cle 9625 . . . . . . 7  class  <_
128, 10, 11wbr 4447 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4999 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  setvar  x
1514cv 1378 . . . . . . . 8  class  x
16 cpnf 9621 . . . . . . . 8  class +oo
17 cico 11527 . . . . . . . 8  class  [,)
1815, 16, 17co 6282 . . . . . . 7  class  ( x [,) +oo )
1913, 18cin 3475 . . . . . 6  class  ( dom  f  i^i  ( x [,) +oo ) )
2012, 2, 19wral 2814 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 9487 . . . . 5  class  RR
2220, 9, 21wrex 2815 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2815 . . 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 9486 . . . 4  class  CC
25 cpm 7418 . . . 4  class  ^pm
2624, 21, 25co 6282 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2818 . 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 1379 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  13308
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