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Theorem isch3 22705
Description: A Hilbert subspace is closed iff it is complete. A complete subspace is one in which every Cauchy sequence of vectors in the subspace converges to a member of the subspace (Definition of complete subspace in [Beran] p. 96). Remark 3.12 of [Beran] p. 107. (Contributed by NM, 24-Dec-2001.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
isch3  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
Distinct variable group:    x, f, H

Proof of Theorem isch3
StepHypRef Expression
1 isch2 22687 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
2 ax-hcompl 22665 . . . . . . . . . 10  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
3 rexex 2733 . . . . . . . . . 10  |-  ( E. x  e.  ~H  f  ~~>v  x  ->  E. x  f  ~~>v  x )
42, 3syl 16 . . . . . . . . 9  |-  ( f  e.  Cauchy  ->  E. x  f  ~~>v  x )
5 19.29 1603 . . . . . . . . 9  |-  ( ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  E. x  f  ~~>v  x )  ->  E. x
( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )
64, 5sylan2 461 . . . . . . . 8  |-  ( ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  e.  Cauchy )  ->  E. x ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )
7 id 20 . . . . . . . . . . . . . . 15  |-  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H
)  ->  ( (
f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
87imp 419 . . . . . . . . . . . . . 14  |-  ( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  (
f : NN --> H  /\  f  ~~>v  x ) )  ->  x  e.  H
)
98an12s 777 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  x  e.  H )
10 simprr 734 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  f  ~~>v  x )
119, 10jca 519 . . . . . . . . . . . 12  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  ( x  e.  H  /\  f  ~~>v  x ) )
1211ex 424 . . . . . . . . . . 11  |-  ( f : NN --> H  -> 
( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( x  e.  H  /\  f  ~~>v  x ) ) )
1312eximdv 1629 . . . . . . . . . 10  |-  ( f : NN --> H  -> 
( E. x ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  E. x
( x  e.  H  /\  f  ~~>v  x ) ) )
1413com12 29 . . . . . . . . 9  |-  ( E. x ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( f : NN --> H  ->  E. x
( x  e.  H  /\  f  ~~>v  x ) ) )
15 df-rex 2680 . . . . . . . . 9  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( x  e.  H  /\  f  ~~>v  x ) )
1614, 15syl6ibr 219 . . . . . . . 8  |-  ( E. x ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
176, 16syl 16 . . . . . . 7  |-  ( ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  e.  Cauchy )  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
1817ex 424 . . . . . 6  |-  ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  ->  ( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
19 nfv 1626 . . . . . . . 8  |-  F/ x  f  e.  Cauchy
20 nfv 1626 . . . . . . . . 9  |-  F/ x  f : NN --> H
21 nfre1 2730 . . . . . . . . 9  |-  F/ x E. x  e.  H  f  ~~>v  x
2220, 21nfim 1828 . . . . . . . 8  |-  F/ x
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )
2319, 22nfim 1828 . . . . . . 7  |-  F/ x
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
24 bi2.04 351 . . . . . . . . 9  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  <-> 
( f : NN --> H  ->  ( f  e. 
Cauchy  ->  E. x  e.  H  f  ~~>v  x ) ) )
25 hlimcaui 22700 . . . . . . . . . . . 12  |-  ( f 
~~>v  x  ->  f  e.  Cauchy )
2625imim1i 56 . . . . . . . . . . 11  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  E. x  e.  H  f  ~~>v  x ) )
27 rexex 2733 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x  f  ~~>v  x )
28 hlimeui 22704 . . . . . . . . . . . . 13  |-  ( E. x  f  ~~>v  x  <->  E! x  f  ~~>v  x )
2927, 28sylib 189 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E! x  f  ~~>v  x )
30 exancom 1593 . . . . . . . . . . . . . 14  |-  ( E. x ( x  e.  H  /\  f  ~~>v  x )  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3115, 30bitri 241 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3231biimpi 187 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x
( f  ~~>v  x  /\  x  e.  H )
)
33 eupick 2325 . . . . . . . . . . . 12  |-  ( ( E! x  f  ~~>v  x  /\  E. x ( f  ~~>v  x  /\  x  e.  H ) )  -> 
( f  ~~>v  x  ->  x  e.  H )
)
3429, 32, 33syl2anc 643 . . . . . . . . . . 11  |-  ( E. x  e.  H  f 
~~>v  x  ->  ( f  ~~>v  x  ->  x  e.  H ) )
3526, 34syli 35 . . . . . . . . . 10  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  x  e.  H ) )
3635imim2i 14 . . . . . . . . 9  |-  ( ( f : NN --> H  -> 
( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3724, 36sylbi 188 . . . . . . . 8  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3837imp3a 421 . . . . . . 7  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
3923, 38alrimi 1777 . . . . . 6  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
4018, 39impbii 181 . . . . 5  |-  ( A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
4140albii 1572 . . . 4  |-  ( A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  A. f ( f  e. 
Cauchy  ->  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
42 df-ral 2679 . . . 4  |-  ( A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )  <->  A. f
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
4341, 42bitr4i 244 . . 3  |-  ( A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  <->  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
4443anbi2i 676 . 2  |-  ( ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H
) )  <->  ( H  e.  SH  /\  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
451, 44bitri 241 1  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f  e.  Cauchy  ( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1721   E!weu 2262   A.wral 2674   E.wrex 2675   class class class wbr 4180   -->wf 5417   NNcn 9964   ~Hchil 22383   Cauchyccau 22390    ~~>v chli 22391   SHcsh 22392   CHcch 22393
This theorem is referenced by:  chcompl  22706  occl  22767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034  ax-hilex 22463  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvmulass 22471  ax-hvdistr1 22472  ax-hvdistr2 22473  ax-hvmul0 22474  ax-hfi 22542  ax-his1 22545  ax-his2 22546  ax-his3 22547  ax-his4 22548  ax-hcompl 22665
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-n0 10186  df-z 10247  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-top 16926  df-bases 16928  df-topon 16929  df-lm 17255  df-haus 17341  df-cau 19170  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040  df-ims 22041  df-hnorm 22432  df-hvsub 22435  df-hlim 22436  df-hcau 22437  df-ch 22685
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