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Theorem h2hlm 26575
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 26567 . . 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 4921 . 2  |-  Rel  ~~>v
3 relres 5094 . 2  |-  Rel  (
( ~~> t `  J
)  |`  ( ~H  ^m  NN ) )
41eleq2i 2498 . . 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 4670 . . 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 451 . . . . 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 26492 . . . . . . 7  |-  ~H  e.  _V
9 nnex 10566 . . . . . . 7  |-  NN  e.  _V
108, 9elmap 7455 . . . . . 6  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
1110anbi1i 699 . . . . 5  |-  ( ( f  e.  ( ~H 
^m  NN )  /\  <.
f ,  x >.  e.  ( ~~> t `  J
) )  <->  ( f : NN --> ~H  /\  <. f ,  x >.  e.  ( ~~> t `  J )
) )
12 df-br 4367 . . . . . . 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 26266 . . . . . . . . . 10  |-  ( U  e.  NrmCVec  ->  D  e.  ( *Met `  ~H ) )
1714, 16mp1i 13 . . . . . . . . 9  |-  ( f : NN --> ~H  ->  D  e.  ( *Met `  ~H ) )
18 nnuz 11145 . . . . . . . . 9  |-  NN  =  ( ZZ>= `  1 )
19 1zzd 10919 . . . . . . . . 9  |-  ( f : NN --> ~H  ->  1  e.  ZZ )
20 eqidd 2429 . . . . . . . . 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 22174 . . . . . . . 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 11180 . . . . . . . . . . . . . 14  |-  ( ( j  e.  NN  /\  k  e.  ( ZZ>= `  j ) )  -> 
k  e.  NN )
24 ffvelrn 5979 . . . . . . . . . . . . . . . . 17  |-  ( ( f : NN --> ~H  /\  k  e.  NN )  ->  ( f `  k
)  e.  ~H )
25 h2hl.1 . . . . . . . . . . . . . . . . . 18  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
2625, 14, 7, 15h2hmetdval 26573 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f `  k
)  e.  ~H  /\  x  e.  ~H )  ->  ( ( f `  k ) D x )  =  ( normh `  ( ( f `  k )  -h  x
) ) )
2724, 26sylan 473 . . . . . . . . . . . . . . . 16  |-  ( ( ( f : NN --> ~H  /\  k  e.  NN )  /\  x  e.  ~H )  ->  ( ( f `
 k ) D x )  =  (
normh `  ( ( f `
 k )  -h  x ) ) )
2827breq1d 4376 . . . . . . . . . . . . . . 15  |-  ( ( ( f : NN --> ~H  /\  k  e.  NN )  /\  x  e.  ~H )  ->  ( ( ( f `  k ) D x )  < 
y  <->  ( normh `  (
( f `  k
)  -h  x ) )  <  y ) )
2928an32s 811 . . . . . . . . . . . . . 14  |-  ( ( ( f : NN --> ~H  /\  x  e.  ~H )  /\  k  e.  NN )  ->  ( ( ( f `  k ) D x )  < 
y  <->  ( normh `  (
( f `  k
)  -h  x ) )  <  y ) )
3023, 29sylan2 476 . . . . . . . . . . . . 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 652 . . . . . . . . . . . 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 2801 . . . . . . . . . . 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 2875 . . . . . . . . . 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 2804 . . . . . . . . 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 645 . . . . . . . 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 256 . . . . . . 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 262 . . . . . 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 641 . . . . 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 275 . . . 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 653 . . . 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 3025 . . . . 5  |-  x  e. 
_V
4241opelres 5072 . . . 4  |-  ( <.
f ,  x >.  e.  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) )  <-> 
( <. f ,  x >.  e.  ( ~~> t `  J )  /\  f  e.  ( ~H  ^m  NN ) ) )
4339, 40, 423bitr4i 280 . . 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 274 . 2  |-  ( <.
f ,  x >.  e. 
~~>v  <->  <. f ,  x >.  e.  ( ( ~~> t `  J )  |`  ( ~H  ^m  NN ) ) )
452, 3, 44eqrelriiv 4891 1  |-  ~~>v  =  ( ( ~~> t `  J
)  |`  ( ~H  ^m  NN ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2714   E.wrex 2715   <.cop 3947   class class class wbr 4366   {copab 4424    |` cres 4798   -->wf 5540   ` cfv 5544  (class class class)co 6249    ^m cmap 7427   1c1 9491    < clt 9626   NNcn 10560   ZZ>=cuz 11110   RR+crp 11253   *Metcxmt 18898   MetOpencmopn 18903   ~~> tclm 20184   NrmCVeccnv 26145   BaseSetcba 26147   IndMetcims 26152   ~Hchil 26514    +h cva 26515    .h csm 26516   normhcno 26518    -h cmv 26520    ~~>v chli 26522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-map 7429  df-pm 7430  df-en 7525  df-dom 7526  df-sdom 7527  df-sup 7909  df-inf 7910  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-div 10221  df-nn 10561  df-2 10619  df-3 10620  df-n0 10821  df-z 10889  df-uz 11111  df-q 11216  df-rp 11254  df-xneg 11360  df-xadd 11361  df-xmul 11362  df-seq 12164  df-exp 12223  df-cj 13106  df-re 13107  df-im 13108  df-sqrt 13242  df-abs 13243  df-topgen 15285  df-psmet 18905  df-xmet 18906  df-met 18907  df-bl 18908  df-mopn 18909  df-top 19863  df-bases 19864  df-topon 19865  df-lm 20187  df-grpo 25861  df-gid 25862  df-ginv 25863  df-gdiv 25864  df-ablo 25952  df-vc 26107  df-nv 26153  df-va 26156  df-ba 26157  df-sm 26158  df-0v 26159  df-vs 26160  df-nmcv 26161  df-ims 26162  df-hvsub 26566  df-hlim 26567
This theorem is referenced by:  axhcompl-zf  26593  hlimadd  26788  hhlm  26794
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