MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elo1 Structured version   Unicode version

Theorem elo1 13004
Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
elo1  |-  ( F  e.  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, m, y, F

Proof of Theorem elo1
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dmeq 5040 . . . . 5  |-  ( f  =  F  ->  dom  f  =  dom  F )
21ineq1d 3551 . . . 4  |-  ( f  =  F  ->  ( dom  f  i^i  (
x [,) +oo )
)  =  ( dom 
F  i^i  ( x [,) +oo ) ) )
3 fveq1 5690 . . . . . 6  |-  ( f  =  F  ->  (
f `  y )  =  ( F `  y ) )
43fveq2d 5695 . . . . 5  |-  ( f  =  F  ->  ( abs `  ( f `  y ) )  =  ( abs `  ( F `  y )
) )
54breq1d 4302 . . . 4  |-  ( f  =  F  ->  (
( abs `  (
f `  y )
)  <_  m  <->  ( abs `  ( F `  y
) )  <_  m
) )
62, 5raleqbidv 2931 . . 3  |-  ( f  =  F  ->  ( A. y  e.  ( dom  f  i^i  (
x [,) +oo )
) ( abs `  (
f `  y )
)  <_  m  <->  A. y  e.  ( dom  F  i^i  ( x [,) +oo ) ) ( abs `  ( F `  y
) )  <_  m
) )
762rexbidv 2758 . 2  |-  ( f  =  F  ->  ( E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  f  i^i  ( x [,) +oo ) ) ( abs `  ( f `  y
) )  <_  m  <->  E. x  e.  RR  E. m  e.  RR  A. y  e.  ( dom  F  i^i  ( x [,) +oo ) ) ( abs `  ( F `  y
) )  <_  m
) )
8 df-o1 12968 . 2  |-  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 }
97, 8elrab2 3119 1  |-  ( F  e.  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
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716    i^i cin 3327   class class class wbr 4292   dom cdm 4840   ` cfv 5418  (class class class)co 6091    ^pm cpm 7215   CCcc 9280   RRcr 9281   +oocpnf 9415    <_ cle 9419   [,)cico 11302   abscabs 12723   O(1)co1 12964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-dm 4850  df-iota 5381  df-fv 5426  df-o1 12968
This theorem is referenced by:  elo12  13005  o1f  13007  o1dm  13008
  Copyright terms: Public domain W3C validator