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Theorem isch3 21814
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 21796 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
2 ax-hcompl 21774 . . . . . . . . . 10  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
3 rexex 2604 . . . . . . . . . 10  |-  ( E. x  e.  ~H  f  ~~>v  x  ->  E. x  f  ~~>v  x )
42, 3syl 17 . . . . . . . . 9  |-  ( f  e.  Cauchy  ->  E. x  f  ~~>v  x )
5 19.29 1584 . . . . . . . . 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 462 . . . . . . . 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 21 . . . . . . . . . . . . . . 15  |-  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H
)  ->  ( (
f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
87imp 420 . . . . . . . . . . . . . 14  |-  ( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  (
f : NN --> H  /\  f  ~~>v  x ) )  ->  x  e.  H
)
98an12s 778 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  x  e.  H )
10 simprr 735 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  f  ~~>v  x )
119, 10jca 520 . . . . . . . . . . . 12  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  ( x  e.  H  /\  f  ~~>v  x ) )
1211ex 425 . . . . . . . . . . 11  |-  ( f : NN --> H  -> 
( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( x  e.  H  /\  f  ~~>v  x ) ) )
1312eximdv 1609 . . . . . . . . . 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 2551 . . . . . . . . 9  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( x  e.  H  /\  f  ~~>v  x ) )
1614, 15syl6ibr 220 . . . . . . . 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 17 . . . . . . 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 425 . . . . . 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 1606 . . . . . . . 8  |-  F/ x  f  e.  Cauchy
20 nfv 1606 . . . . . . . . 9  |-  F/ x  f : NN --> H
21 nfre1 2601 . . . . . . . . 9  |-  F/ x E. x  e.  H  f  ~~>v  x
2220, 21nfim 1771 . . . . . . . 8  |-  F/ x
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )
2319, 22nfim 1771 . . . . . . 7  |-  F/ x
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
24 bi2.04 352 . . . . . . . . 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 21809 . . . . . . . . . . . 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 2604 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x  f  ~~>v  x )
28 hlimeui 21813 . . . . . . . . . . . . 13  |-  ( E. x  f  ~~>v  x  <->  E! x  f  ~~>v  x )
2927, 28sylib 190 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E! x  f  ~~>v  x )
30 exancom 1574 . . . . . . . . . . . . . 14  |-  ( E. x ( x  e.  H  /\  f  ~~>v  x )  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3115, 30bitri 242 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3231biimpi 188 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x
( f  ~~>v  x  /\  x  e.  H )
)
33 eupick 2208 . . . . . . . . . . . 12  |-  ( ( E! x  f  ~~>v  x  /\  E. x ( f  ~~>v  x  /\  x  e.  H ) )  -> 
( f  ~~>v  x  ->  x  e.  H )
)
3429, 32, 33syl2anc 644 . . . . . . . . . . 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 15 . . . . . . . . 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 189 . . . . . . . 8  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3837imp3a 422 . . . . . . 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 1747 . . . . . 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 182 . . . . 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 1554 . . . 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 2550 . . . 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 245 . . 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 677 . 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 242 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 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    e. wcel 1685   E!weu 2145   A.wral 2545   E.wrex 2546   class class class wbr 4025   -->wf 5218   NNcn 9742   ~Hchil 21492   Cauchyccau 21499    ~~>v chli 21500   SHcsh 21501   CHcch 21502
This theorem is referenced by:  chcompl  21815  occl  21876
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813  ax-hilex 21572  ax-hfvadd 21573  ax-hvcom 21574  ax-hvass 21575  ax-hv0cl 21576  ax-hvaddid 21577  ax-hfvmul 21578  ax-hvmulid 21579  ax-hvmulass 21580  ax-hvdistr1 21581  ax-hvdistr2 21582  ax-hvmul0 21583  ax-hfi 21651  ax-his1 21654  ax-his2 21655  ax-his3 21656  ax-his4 21657  ax-hcompl 21774
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-er 6656  df-map 6770  df-pm 6771  df-en 6860  df-dom 6861  df-sdom 6862  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-n0 9962  df-z 10021  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-icc 10658  df-seq 11042  df-exp 11100  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-topgen 13339  df-xmet 16368  df-met 16369  df-bl 16370  df-mopn 16371  df-top 16631  df-bases 16633  df-topon 16634  df-lm 16954  df-haus 17038  df-cau 18677  df-grpo 20851  df-gid 20852  df-ginv 20853  df-gdiv 20854  df-ablo 20942  df-vc 21095  df-nv 21141  df-va 21144  df-ba 21145  df-sm 21146  df-0v 21147  df-vs 21148  df-nmcv 21149  df-ims 21150  df-hnorm 21541  df-hvsub 21544  df-hlim 21545  df-hcau 21546  df-ch 21794
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