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Theorem hlimi 21767
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 21552 . . . 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 4811 . . 3  |-  Rel  ~~>v
32brrelexi 4729 . 2  |-  ( F 
~~>v  A  ->  F  e.  _V )
4 nnex 9752 . . . 4  |-  NN  e.  _V
5 fex 5749 . . . 4  |-  ( ( F : NN --> ~H  /\  NN  e.  _V )  ->  F  e.  _V )
64, 5mpan2 652 . . 3  |-  ( F : NN --> ~H  ->  F  e.  _V )
76ad2antrr 706 . 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 5375 . . . . . 6  |-  ( f  =  F  ->  (
f : NN --> ~H  <->  F : NN
--> ~H ) )
10 eleq1 2343 . . . . . 6  |-  ( w  =  A  ->  (
w  e.  ~H  <->  A  e.  ~H ) )
119, 10bi2anan9 843 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f : NN --> ~H  /\  w  e.  ~H )  <->  ( F : NN --> ~H  /\  A  e. 
~H ) ) )
12 fveq1 5524 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
13 oveq12 5867 . . . . . . . . . 10  |-  ( ( ( f `  z
)  =  ( F `
 z )  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1412, 13sylan 457 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1514fveq2d 5529 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  A )  ->  ( normh `  ( (
f `  z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
1615breq1d 4033 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( normh `  (
( f `  z
)  -h  w ) )  <  x  <->  ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1716rexralbidv 2587 . . . . . 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 2563 . . . . 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 691 . . . 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 4279 . . 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 652 . 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 342 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858    < clt 8867   NNcn 9746   ZZ>=cuz 10230   RR+crp 10354   ~Hchil 21499   normhcno 21503    -h cmv 21505    ~~>v chli 21507
This theorem is referenced by:  hlimseqi  21768  hlimveci  21769  hlimconvi  21770  hlim2  21771
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-nn 9747  df-hlim 21552
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