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Definition df-o1 13398
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 13394 . 2  class  O(1)
2 vy . . . . . . . . . 10  setvar  y
32cv 1397 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  setvar  f
54cv 1397 . . . . . . . . 9  class  f
63, 5cfv 5570 . . . . . . . 8  class  ( f `
 y )
7 cabs 13152 . . . . . . . 8  class  abs
86, 7cfv 5570 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  setvar  m
109cv 1397 . . . . . . 7  class  m
11 cle 9618 . . . . . . 7  class  <_
128, 10, 11wbr 4439 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4988 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  setvar  x
1514cv 1397 . . . . . . . 8  class  x
16 cpnf 9614 . . . . . . . 8  class +oo
17 cico 11534 . . . . . . . 8  class  [,)
1815, 16, 17co 6270 . . . . . . 7  class  ( x [,) +oo )
1913, 18cin 3460 . . . . . 6  class  ( dom  f  i^i  ( x [,) +oo ) )
2012, 2, 19wral 2804 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 9480 . . . . 5  class  RR
2220, 9, 21wrex 2805 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2805 . . 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 9479 . . . 4  class  CC
25 cpm 7413 . . . 4  class  ^pm
2624, 21, 25co 6270 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2808 . 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 1398 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  13434
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