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Theorem isch3 24643
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 24625 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
2 ax-hcompl 24603 . . . . . . . . . 10  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
3 rexex 2774 . . . . . . . . . 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 1650 . . . . . . . . 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 474 . . . . . . . 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 22 . . . . . . . . . . . . . . 15  |-  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H
)  ->  ( (
f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
87imp 429 . . . . . . . . . . . . . 14  |-  ( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  (
f : NN --> H  /\  f  ~~>v  x ) )  ->  x  e.  H
)
98an12s 799 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  x  e.  H )
10 simprr 756 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  f  ~~>v  x )
119, 10jca 532 . . . . . . . . . . . 12  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  ( x  e.  H  /\  f  ~~>v  x ) )
1211ex 434 . . . . . . . . . . 11  |-  ( f : NN --> H  -> 
( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( x  e.  H  /\  f  ~~>v  x ) ) )
1312eximdv 1676 . . . . . . . . . 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 31 . . . . . . . . 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 2720 . . . . . . . . 9  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( x  e.  H  /\  f  ~~>v  x ) )
1614, 15syl6ibr 227 . . . . . . . 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 434 . . . . . 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 1673 . . . . . . . 8  |-  F/ x  f  e.  Cauchy
20 nfv 1673 . . . . . . . . 9  |-  F/ x  f : NN --> H
21 nfre1 2771 . . . . . . . . 9  |-  F/ x E. x  e.  H  f  ~~>v  x
2220, 21nfim 1853 . . . . . . . 8  |-  F/ x
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )
2319, 22nfim 1853 . . . . . . 7  |-  F/ x
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
24 bi2.04 361 . . . . . . . . 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 24638 . . . . . . . . . . . 12  |-  ( f 
~~>v  x  ->  f  e.  Cauchy )
2625imim1i 58 . . . . . . . . . . 11  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  E. x  e.  H  f  ~~>v  x ) )
27 rexex 2774 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x  f  ~~>v  x )
28 hlimeui 24642 . . . . . . . . . . . . 13  |-  ( E. x  f  ~~>v  x  <->  E! x  f  ~~>v  x )
2927, 28sylib 196 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E! x  f  ~~>v  x )
30 exancom 1638 . . . . . . . . . . . . 13  |-  ( E. x ( x  e.  H  /\  f  ~~>v  x )  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3115, 30sylbb 197 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x
( f  ~~>v  x  /\  x  e.  H )
)
32 eupick 2342 . . . . . . . . . . . 12  |-  ( ( E! x  f  ~~>v  x  /\  E. x ( f  ~~>v  x  /\  x  e.  H ) )  -> 
( f  ~~>v  x  ->  x  e.  H )
)
3329, 31, 32syl2anc 661 . . . . . . . . . . 11  |-  ( E. x  e.  H  f 
~~>v  x  ->  ( f  ~~>v  x  ->  x  e.  H ) )
3426, 33syli 37 . . . . . . . . . 10  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  x  e.  H ) )
3534imim2i 14 . . . . . . . . 9  |-  ( ( f : NN --> H  -> 
( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3624, 35sylbi 195 . . . . . . . 8  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3736impd 431 . . . . . . 7  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
)
3823, 37alrimi 1811 . . . . . 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 )
)
3918, 38impbii 188 . . . . 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 ) ) )
4039albii 1610 . . . 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 ) ) )
41 df-ral 2719 . . . 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 ) ) )
4240, 41bitr4i 252 . . 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 ) )
4342anbi2i 694 . 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 ) ) )
441, 43bitri 249 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367   E.wex 1586    e. wcel 1756   E!weu 2253   A.wral 2714   E.wrex 2715   class class class wbr 4291   -->wf 5413   NNcn 10321   ~Hchil 24320   Cauchyccau 24327    ~~>v chli 24328   SHcsh 24329   CHcch 24330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358  ax-pre-sup 9359  ax-addf 9360  ax-mulf 9361  ax-hilex 24400  ax-hfvadd 24401  ax-hvcom 24402  ax-hvass 24403  ax-hv0cl 24404  ax-hvaddid 24405  ax-hfvmul 24406  ax-hvmulid 24407  ax-hvmulass 24408  ax-hvdistr1 24409  ax-hvdistr2 24410  ax-hvmul0 24411  ax-hfi 24480  ax-his1 24483  ax-his2 24484  ax-his3 24485  ax-his4 24486  ax-hcompl 24603
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-map 7215  df-pm 7216  df-en 7310  df-dom 7311  df-sdom 7312  df-sup 7690  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-div 9993  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-n0 10579  df-z 10646  df-uz 10861  df-q 10953  df-rp 10991  df-xneg 11088  df-xadd 11089  df-xmul 11090  df-icc 11306  df-seq 11806  df-exp 11865  df-cj 12587  df-re 12588  df-im 12589  df-sqr 12723  df-abs 12724  df-topgen 14381  df-psmet 17808  df-xmet 17809  df-met 17810  df-bl 17811  df-mopn 17812  df-top 18502  df-bases 18504  df-topon 18505  df-lm 18832  df-haus 18918  df-cau 20766  df-grpo 23677  df-gid 23678  df-ginv 23679  df-gdiv 23680  df-ablo 23768  df-vc 23923  df-nv 23969  df-va 23972  df-ba 23973  df-sm 23974  df-0v 23975  df-vs 23976  df-nmcv 23977  df-ims 23978  df-hnorm 24369  df-hvsub 24372  df-hlim 24373  df-hcau 24374  df-ch 24623
This theorem is referenced by:  chcompl  24644  occl  24706
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