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Theorem hlimi 24595
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 24379 . . . 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 4970 . . 3  |-  Rel  ~~>v
32brrelexi 4884 . 2  |-  ( F 
~~>v  A  ->  F  e.  _V )
4 nnex 10333 . . . 4  |-  NN  e.  _V
5 fex 5955 . . . 4  |-  ( ( F : NN --> ~H  /\  NN  e.  _V )  ->  F  e.  _V )
64, 5mpan2 671 . . 3  |-  ( F : NN --> ~H  ->  F  e.  _V )
76ad2antrr 725 . 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 5547 . . . . . 6  |-  ( f  =  F  ->  (
f : NN --> ~H  <->  F : NN
--> ~H ) )
10 eleq1 2503 . . . . . 6  |-  ( w  =  A  ->  (
w  e.  ~H  <->  A  e.  ~H ) )
119, 10bi2anan9 868 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f : NN --> ~H  /\  w  e.  ~H )  <->  ( F : NN --> ~H  /\  A  e. 
~H ) ) )
12 fveq1 5695 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
13 oveq12 6105 . . . . . . . . . 10  |-  ( ( ( f `  z
)  =  ( F `
 z )  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1412, 13sylan 471 . . . . . . . . 9  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f `  z )  -h  w
)  =  ( ( F `  z )  -h  A ) )
1514fveq2d 5700 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  A )  ->  ( normh `  ( (
f `  z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
1615breq1d 4307 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( normh `  (
( f `  z
)  -h  w ) )  <  x  <->  ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1716rexralbidv 2764 . . . . . 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 2740 . . . . 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 710 . . . 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 4608 . . 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 671 . 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 353 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   _Vcvv 2977   class class class wbr 4297   -->wf 5419   ` cfv 5423  (class class class)co 6096    < clt 9423   NNcn 10327   ZZ>=cuz 10866   RR+crp 10996   ~Hchil 24326   normhcno 24330    -h cmv 24332    ~~>v chli 24334
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-i2m1 9355  ax-1ne0 9356  ax-rrecex 9359  ax-cnre 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-om 6482  df-recs 6837  df-rdg 6871  df-nn 10328  df-hlim 24379
This theorem is referenced by:  hlimseqi  24596  hlimveci  24597  hlimconvi  24598  hlim2  24599
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