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

Theorem hlimi 21727
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
StepHypRef Expression
1 df-hlim 21512 . . . 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 4799 . . 3  |-  Rel  ~~>v
32brrelexi 4717 . 2  |-  ( F 
~~>v  A  ->  F  e.  _V )
4 nnex 9720 . . . 4  |-  NN  e.  _V
5 fex 5683 . . . 4  |-  ( ( F : NN --> ~H  /\  NN  e.  _V )  ->  F  e.  _V )
64, 5mpan2 655 . . 3  |-  ( F : NN --> ~H  ->  F  e.  _V )
76ad2antrr 709 . 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 5313 . . . . . 6  |-  ( f  =  F  ->  (
f : NN --> ~H  <->  F : NN
--> ~H ) )
10 eleq1 2318 . . . . . 6  |-  ( w  =  A  ->  (
w  e.  ~H  <->  A  e.  ~H ) )
119, 10bi2anan9 848 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f : NN --> ~H  /\  w  e.  ~H )  <->  ( F : NN --> ~H  /\  A  e. 
~H ) ) )
12 fveq1 5457 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
13 oveq12 5801 . . . . . . . . . 10  |-  ( ( ( f `  z
)  =  ( F `
 z )  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1412, 13sylan 459 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1514fveq2d 5462 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  A )  ->  ( normh `  ( (
f `  z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
1615breq1d 4007 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( normh `  (
( f `  z
)  -h  w ) )  <  x  <->  ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1716rexralbidv 2562 . . . . . 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 2538 . . . . 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 694 . . . 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 4251 . . 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 655 . 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 344 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 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518   E.wrex 2519   _Vcvv 2763   class class class wbr 3997   -->wf 4669   ` cfv 4673  (class class class)co 5792    < clt 8835   NNcn 9714   ZZ>=cuz 10197   RR+crp 10321   ~Hchil 21459   normhcno 21463    -h cmv 21465    ~~>v chli 21467
This theorem is referenced by:  hlimseqi  21728  hlimveci  21729  hlimconvi  21730  hlim2  21731
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-i2m1 8773  ax-1ne0 8774  ax-rrecex 8777  ax-cnre 8778
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-recs 6356  df-rdg 6391  df-n 9715  df-hlim 21512
  Copyright terms: Public domain W3C validator