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Theorem hlimi 26231
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 26015 . . . 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 5137 . . 3  |-  Rel  ~~>v
32brrelexi 5049 . 2  |-  ( F 
~~>v  A  ->  F  e.  _V )
4 nnex 10562 . . . 4  |-  NN  e.  _V
5 fex 6146 . . . 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 5719 . . . . . 6  |-  ( f  =  F  ->  (
f : NN --> ~H  <->  F : NN
--> ~H ) )
10 eleq1 2529 . . . . . 6  |-  ( w  =  A  ->  (
w  e.  ~H  <->  A  e.  ~H ) )
119, 10bi2anan9 873 . . . . 5  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( f : NN --> ~H  /\  w  e.  ~H )  <->  ( F : NN --> ~H  /\  A  e. 
~H ) ) )
12 fveq1 5871 . . . . . . . . . 10  |-  ( f  =  F  ->  (
f `  z )  =  ( F `  z ) )
13 oveq12 6305 . . . . . . . . . 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 5876 . . . . . . . 8  |-  ( ( f  =  F  /\  w  =  A )  ->  ( normh `  ( (
f `  z )  -h  w ) )  =  ( normh `  ( ( F `  z )  -h  A ) ) )
1615breq1d 4466 . . . . . . 7  |-  ( ( f  =  F  /\  w  =  A )  ->  ( ( normh `  (
( f `  z
)  -h  w ) )  <  x  <->  ( normh `  ( ( F `  z )  -h  A
) )  <  x
) )
1716rexralbidv 2976 . . . . . 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 2896 . . . . 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 4770 . . 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 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109   class class class wbr 4456   -->wf 5590   ` cfv 5594  (class class class)co 6296    < clt 9645   NNcn 10556   ZZ>=cuz 11106   RR+crp 11245   ~Hchil 25962   normhcno 25966    -h cmv 25968    ~~>v chli 25970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rrecex 9581  ax-cnre 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-om 6700  df-recs 7060  df-rdg 7094  df-nn 10557  df-hlim 26015
This theorem is referenced by:  hlimseqi  26232  hlimveci  26233  hlimconvi  26234  hlim2  26235
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