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

Definition df-o1 12973
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 12969 . 2  class  O(1)
2 vy . . . . . . . . . 10  setvar  y
32cv 1368 . . . . . . . . 9  class  y
4 vf . . . . . . . . . 10  setvar  f
54cv 1368 . . . . . . . . 9  class  f
63, 5cfv 5423 . . . . . . . 8  class  ( f `
 y )
7 cabs 12728 . . . . . . . 8  class  abs
86, 7cfv 5423 . . . . . . 7  class  ( abs `  ( f `  y
) )
9 vm . . . . . . . 8  setvar  m
109cv 1368 . . . . . . 7  class  m
11 cle 9424 . . . . . . 7  class  <_
128, 10, 11wbr 4297 . . . . . 6  wff  ( abs `  ( f `  y
) )  <_  m
135cdm 4845 . . . . . . 7  class  dom  f
14 vx . . . . . . . . 9  setvar  x
1514cv 1368 . . . . . . . 8  class  x
16 cpnf 9420 . . . . . . . 8  class +oo
17 cico 11307 . . . . . . . 8  class  [,)
1815, 16, 17co 6096 . . . . . . 7  class  ( x [,) +oo )
1913, 18cin 3332 . . . . . 6  class  ( dom  f  i^i  ( x [,) +oo ) )
2012, 2, 19wral 2720 . . . . 5  wff  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
21 cr 9286 . . . . 5  class  RR
2220, 9, 21wrex 2721 . . . 4  wff  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m
2322, 14, 21wrex 2721 . . 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 9285 . . . 4  class  CC
25 cpm 7220 . . . 4  class  ^pm
2624, 21, 25co 6096 . . 3  class  ( CC 
^pm  RR )
2723, 4, 26crab 2724 . 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 1369 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  13009
  Copyright terms: Public domain W3C validator