HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hlim2 Unicode version

Theorem hlim2 22655
Description: The limit of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hlim2  |-  ( ( F : NN --> ~H  /\  A  e.  ~H )  ->  ( F  ~~>v  A  <->  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 hlim2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 breq2 4184 . . . . 5  |-  ( w  =  A  ->  ( F  ~~>v  w  <->  F  ~~>v  A ) )
2 oveq2 6056 . . . . . . . . 9  |-  ( w  =  A  ->  (
( F `  z
)  -h  w )  =  ( ( F `
 z )  -h  A ) )
32fveq2d 5699 . . . . . . . 8  |-  ( w  =  A  ->  ( normh `  ( ( F `
 z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
43breq1d 4190 . . . . . . 7  |-  ( w  =  A  ->  (
( normh `  ( ( F `  z )  -h  w ) )  < 
x  <->  ( normh `  (
( F `  z
)  -h  A ) )  <  x ) )
54rexralbidv 2718 . . . . . 6  |-  ( 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
) )
65ralbidv 2694 . . . . 5  |-  ( 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
) )
71, 6bibi12d 313 . . . 4  |-  ( w  =  A  ->  (
( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x )  <->  ( F  ~~>v  A 
<-> 
A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) )
87imbi2d 308 . . 3  |-  ( w  =  A  ->  (
( F : NN --> ~H  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) )  <->  ( F : NN --> ~H  ->  ( F 
~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y ) ( normh `  ( ( F `  z )  -h  A
) )  <  x
) ) ) )
9 vex 2927 . . . . . 6  |-  w  e. 
_V
109hlimi 22651 . . . . 5  |-  ( F 
~~>v  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
) )
1110baib 872 . . . 4  |-  ( ( F : NN --> ~H  /\  w  e.  ~H )  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) )
1211expcom 425 . . 3  |-  ( w  e.  ~H  ->  ( F : NN --> ~H  ->  ( F  ~~>v  w  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  w ) )  < 
x ) ) )
138, 12vtoclga 2985 . 2  |-  ( A  e.  ~H  ->  ( F : NN --> ~H  ->  ( F  ~~>v  A  <->  A. x  e.  RR+  E. y  e.  NN  A. z  e.  ( ZZ>= `  y )
( normh `  ( ( F `  z )  -h  A ) )  < 
x ) ) )
1413impcom 420 1  |-  ( ( F : NN --> ~H  /\  A  e.  ~H )  ->  ( F  ~~>v  A  <->  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:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   class class class wbr 4180   -->wf 5417   ` cfv 5421  (class class class)co 6048    < clt 9084   NNcn 9964   ZZ>=cuz 10452   RR+crp 10576   ~Hchil 22383   normhcno 22387    -h cmv 22389    ~~>v chli 22391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rrecex 9026  ax-cnre 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-recs 6600  df-rdg 6635  df-nn 9965  df-hlim 22436
  Copyright terms: Public domain W3C validator