MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-o1 Unicode version

Definition df-o1 12239
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 12235 . 2  class  O ( 1 )
2 vy . . . . . . . . . 10  set  y
32cv 1648 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  set  f
54cv 1648 . . . . . . . . 9  class  f
63, 5cfv 5413 . . . . . . . 8  class  ( f `
 y )
7 cabs 11994 . . . . . . . 8  class  abs
86, 7cfv 5413 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  set  m
109cv 1648 . . . . . . 7  class  m
11 cle 9077 . . . . . . 7  class  <_
128, 10, 11wbr 4172 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4837 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  set  x
1514cv 1648 . . . . . . . 8  class  x
16 cpnf 9073 . . . . . . . 8  class  +oo
17 cico 10874 . . . . . . . 8  class  [,)
1815, 16, 17co 6040 . . . . . . 7  class  ( x [,)  +oo )
1913, 18cin 3279 . . . . . 6  class  ( dom  f  i^i  ( x [,)  +oo ) )
2012, 2, 19wral 2666 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 8945 . . . . 5  class  RR
2220, 9, 21wrex 2667 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,)  +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2667 . . 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 8944 . . . 4  class  CC
25 cpm 6978 . . . 4  class  ^pm
2624, 21, 25co 6040 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2670 . 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 1649 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  12275
  Copyright terms: Public domain W3C validator