HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hlimi Unicode version

Theorem hlimi 22678
Description: Express the predicate: The limit of vector sequence  F in a Hilbert space is  A, i.e.  F converges to  A. This means that for any real  x, no matter how small, there always exists an integer  y such that the norm of any later vector in the sequence minus the limit is less than  x. Definition of converge in [Beran] p. 96. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlim.1  |-  A  e. 
_V
Assertion
Ref Expression
hlimi  |-  ( F 
~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
Distinct variable groups:    x, y,
z, F    x, A, y, z

Proof of Theorem hlimi
Dummy variables  w  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hlim 22463 . . . 4  |-  ~~>v  =  { <. f ,  w >.  |  ( ( f : NN --> ~H  /\  w  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( (
f `  z )  -h  w ) )  < 
x ) }
21relopabi 4991 . . 3  |-  Rel  ~~>v
32brrelexi 4909 . 2  |-  ( F 
~~>v  A  ->  F  e.  _V )
4 nnex 9995 . . . 4  |-  NN  e.  _V
5 fex 5960 . . . 4  |-  ( ( F : NN --> ~H  /\  NN  e.  _V )  ->  F  e.  _V )
64, 5mpan2 653 . . 3  |-  ( F : NN --> ~H  ->  F  e.  _V )
76ad2antrr 707 . 2  |-  ( ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
)  ->  F  e.  _V )
8 hlim.1 . . 3  |-  A  e. 
_V
9 feq1 5567 . . . . . 6  |-  ( f  =  F  ->  (
f : NN --> ~H  <->  F : NN
--> ~H ) )
10 eleq1 2495 . . . . . 6  |-  ( w  =  A  ->  (
w  e.  ~H  <->  A  e.  ~H ) )
119, 10bi2anan9 844 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f : NN --> ~H  /\  w  e.  ~H )  <->  ( F : NN --> ~H  /\  A  e. 
~H ) ) )
12 fveq1 5718 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
13 oveq12 6081 . . . . . . . . . 10  |-  ( ( ( f `  z
)  =  ( F `
 z )  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1412, 13sylan 458 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1514fveq2d 5723 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  A )  ->  ( normh `  ( (
f `  z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
1615breq1d 4214 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( normh `  (
( f `  z
)  -h  w ) )  <  x  <->  ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1716rexralbidv 2741 . . . . . 6  |-  ( ( f  =  F  /\  w  =  A )  ->  ( E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( (
f `  z )  -h  w ) )  < 
x  <->  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1817ralbidv 2717 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( f `  z )  -h  w
) )  <  x  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  A ) )  < 
x ) )
1911, 18anbi12d 692 . . . 4  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( ( f : NN --> ~H  /\  w  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( f `  z )  -h  w
) )  <  x
)  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) )
2019, 1brabga 4461 . . 3  |-  ( ( F  e.  _V  /\  A  e.  _V )  ->  ( F  ~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) )
218, 20mpan2 653 . 2  |-  ( F  e.  _V  ->  ( F  ~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) )
223, 7, 21pm5.21nii 343 1  |-  ( F 
~~>v  A  <->  ( ( F : NN --> ~H  /\  A  e.  ~H )  /\  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   _Vcvv 2948   class class class wbr 4204   -->wf 5441   ` cfv 5445  (class class class)co 6072    < clt 9109   NNcn 9989   ZZ>=cuz 10477   RR+crp 10601   ~Hchil 22410   normhcno 22414    -h cmv 22416    ~~>v chli 22418
This theorem is referenced by:  hlimseqi  22679  hlimveci  22680  hlimconvi  22681  hlim2  22682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-i2m1 9047  ax-1ne0 9048  ax-rrecex 9051  ax-cnre 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-recs 6624  df-rdg 6659  df-nn 9990  df-hlim 22463
  Copyright terms: Public domain W3C validator