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Theorem h2hlm 26714
Description: The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2hl.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
h2hl.2  |-  U  e.  NrmCVec
h2hl.3  |-  ~H  =  ( BaseSet `  U )
h2hl.4  |-  D  =  ( IndMet `  U )
h2hl.5  |-  J  =  ( MetOpen `  D )
Assertion
Ref Expression
h2hlm  |-  ~~>v  =  ( ( ~~> t `  J
)  |`  ( ~H  ^m  NN ) )

Proof of Theorem h2hlm
Dummy variables  x  f  y  j  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-hlim 26706 . . 3  |-  ~~>v  =  { <. f ,  x >.  |  ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) }
21relopabi 4964 . 2  |-  Rel  ~~>v
3 relres 5138 . 2  |-  Rel  (
( ~~> t `  J
)  |`  ( ~H  ^m  NN ) )
41eleq2i 2541 . . 3  |-  ( <.
f ,  x >.  e. 
~~>v  <->  <. f ,  x >.  e. 
{ <. f ,  x >.  |  ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
) } )
5 opabid 4708 . . 3  |-  ( <.
f ,  x >.  e. 
{ <. f ,  x >.  |  ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
) }  <->  ( (
f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
) )
6 ancom 457 . . . . 5  |-  ( (
<. f ,  x >.  e.  ( ~~> t `  J
)  /\  f  e.  ( ~H  ^m  NN ) )  <->  ( f  e.  ( ~H  ^m  NN )  /\  <. f ,  x >.  e.  ( ~~> t `  J ) ) )
7 h2hl.3 . . . . . . . 8  |-  ~H  =  ( BaseSet `  U )
87hlex 26631 . . . . . . 7  |-  ~H  e.  _V
9 nnex 10637 . . . . . . 7  |-  NN  e.  _V
108, 9elmap 7518 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
1110anbi1i 709 . . . . 5  |-  ( ( f  e.  ( ~H 
^m  NN )  /\  <.
f ,  x >.  e.  ( ~~> t `  J
) )  <->  ( f : NN --> ~H  /\  <. f ,  x >.  e.  ( ~~> t `  J )
) )
12 df-br 4396 . . . . . . 7  |-  ( f ( ~~> t `  J
) x  <->  <. f ,  x >.  e.  ( ~~> t `  J )
)
13 h2hl.5 . . . . . . . . 9  |-  J  =  ( MetOpen `  D )
14 h2hl.2 . . . . . . . . . 10  |-  U  e.  NrmCVec
15 h2hl.4 . . . . . . . . . . 11  |-  D  =  ( IndMet `  U )
167, 15imsxmet 26405 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  D  e.  ( *Met `  ~H ) )
1714, 16mp1i 13 . . . . . . . . 9  |-  ( f : NN --> ~H  ->  D  e.  ( *Met `  ~H ) )
18 nnuz 11218 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
19 1zzd 10992 . . . . . . . . 9  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
20 eqidd 2472 . . . . . . . . 9  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  =  ( f `
 k ) )
21 id 22 . . . . . . . . 9  |-  ( f : NN --> ~H  ->  f : NN --> ~H )
2213, 17, 18, 19, 20, 21lmmbrf 22310 . . . . . . . 8  |-  ( f : NN --> ~H  ->  ( f ( ~~> t `  J ) x  <->  ( x  e.  ~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( f `  k ) D x )  <  y ) ) )
23 eluznn 11252 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
24 ffvelrn 6035 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
25 h2hl.1 . . . . . . . . . . . . . . . . . 18  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
2625, 14, 7, 15h2hmetdval 26712 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f `  k
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( f `  k ) D x )  =  ( normh `  ( ( f `  k )  -h  x
) ) )
2724, 26sylan 479 . . . . . . . . . . . . . . . 16  |-  ( ( ( f : NN --> ~H  /\  k  e.  NN )  /\  x  e.  ~H )  ->  ( ( f `
 k ) D x )  =  (
normh `  ( ( f `
 k )  -h  x ) ) )
2827breq1d 4405 . . . . . . . . . . . . . . 15  |-  ( ( ( f : NN --> ~H  /\  k  e.  NN )  /\  x  e.  ~H )  ->  ( ( ( f `  k ) D x )  < 
y  <->  ( normh `  (
( f `  k
)  -h  x ) )  <  y ) )
2928an32s 821 . . . . . . . . . . . . . 14  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  k  e.  NN )  ->  ( ( ( f `  k ) D x )  < 
y  <->  ( normh `  (
( f `  k
)  -h  x ) )  <  y ) )
3023, 29sylan2 482 . . . . . . . . . . . . 13  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  ( j  e.  NN  /\  k  e.  ( ZZ>= `  j )
) )  ->  (
( ( f `  k ) D x )  <  y  <->  ( normh `  ( ( f `  k )  -h  x
) )  <  y
) )
3130anassrs 660 . . . . . . . . . . . 12  |-  ( ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  j  e.  NN )  /\  k  e.  ( ZZ>= `  j )
)  ->  ( (
( f `  k
) D x )  <  y  <->  ( normh `  ( ( f `  k )  -h  x
) )  <  y
) )
3231ralbidva 2828 . . . . . . . . . . 11  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  j  e.  NN )  ->  ( A. k  e.  ( ZZ>= `  j )
( ( f `  k ) D x )  <  y  <->  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) )
3332rexbidva 2889 . . . . . . . . . 10  |-  ( ( f : NN --> ~H  /\  x  e.  ~H )  ->  ( E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( ( f `  k ) D x )  <  y  <->  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) )
3433ralbidv 2829 . . . . . . . . 9  |-  ( ( f : NN --> ~H  /\  x  e.  ~H )  ->  ( A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( f `  k ) D x )  < 
y  <->  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
) )
3534pm5.32da 653 . . . . . . . 8  |-  ( f : NN --> ~H  ->  ( ( x  e.  ~H  /\ 
A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( ( f `  k ) D x )  < 
y )  <->  ( x  e.  ~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) ) )
3622, 35bitrd 261 . . . . . . 7  |-  ( f : NN --> ~H  ->  ( f ( ~~> t `  J ) x  <->  ( x  e.  ~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) ) )
3712, 36syl5bbr 267 . . . . . 6  |-  ( f : NN --> ~H  ->  (
<. f ,  x >.  e.  ( ~~> t `  J
)  <->  ( x  e. 
~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) ) )
3837pm5.32i 649 . . . . 5  |-  ( ( f : NN --> ~H  /\  <.
f ,  x >.  e.  ( ~~> t `  J
) )  <->  ( f : NN --> ~H  /\  (
x  e.  ~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) ) )
396, 11, 383bitrri 280 . . . 4  |-  ( ( f : NN --> ~H  /\  ( x  e.  ~H  /\ 
A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
) )  <->  ( <. f ,  x >.  e.  ( ~~> t `  J )  /\  f  e.  ( ~H  ^m  NN ) ) )
40 anass 661 . . . 4  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
)  <->  ( f : NN --> ~H  /\  (
x  e.  ~H  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j )
( normh `  ( (
f `  k )  -h  x ) )  < 
y ) ) )
41 vex 3034 . . . . 5  |-  x  e. 
_V
4241opelres 5116 . . . 4  |-  ( <.
f ,  x >.  e.  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) )  <-> 
( <. f ,  x >.  e.  ( ~~> t `  J )  /\  f  e.  ( ~H  ^m  NN ) ) )
4339, 40, 423bitr4i 285 . . 3  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  A. y  e.  RR+  E. j  e.  NN  A. k  e.  ( ZZ>= `  j ) ( normh `  ( ( f `  k )  -h  x
) )  <  y
)  <->  <. f ,  x >.  e.  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) ) )
444, 5, 433bitri 279 . 2  |-  ( <.
f ,  x >.  e. 
~~>v  <->  <. f ,  x >.  e.  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) ) )
452, 3, 44eqrelriiv 4934 1  |-  ~~>v  =  ( ( ~~> t `  J
)  |`  ( ~H  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   <.cop 3965   class class class wbr 4395   {copab 4453    |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   1c1 9558    < clt 9693   NNcn 10631   ZZ>=cuz 11182   RR+crp 11325   *Metcxmt 19032   MetOpencmopn 19037   ~~> tclm 20319   NrmCVeccnv 26284   BaseSetcba 26286   IndMetcims 26291   ~Hchil 26653    +h cva 26654    .h csm 26655   normhcno 26657    -h cmv 26659    ~~>v chli 26661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-lm 20322  df-grpo 26000  df-gid 26001  df-ginv 26002  df-gdiv 26003  df-ablo 26091  df-vc 26246  df-nv 26292  df-va 26295  df-ba 26296  df-sm 26297  df-0v 26298  df-vs 26299  df-nmcv 26300  df-ims 26301  df-hvsub 26705  df-hlim 26706
This theorem is referenced by:  axhcompl-zf  26732  hlimadd  26927  hhlm  26933
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