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Theorem isch3 26975
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 26957 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
2 ax-hcompl 26936 . . . . . . . . . 10  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
3 rexex 2843 . . . . . . . . . 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 1743 . . . . . . . . 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 482 . . . . . . . 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 436 . . . . . . . . . . . . . 14  |-  ( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  (
f : NN --> H  /\  f  ~~>v  x ) )  ->  x  e.  H
)
98an12s 818 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  x  e.  H )
10 simprr 774 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  f  ~~>v  x )
119, 10jca 541 . . . . . . . . . . . 12  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  ( x  e.  H  /\  f  ~~>v  x ) )
1211ex 441 . . . . . . . . . . 11  |-  ( f : NN --> H  -> 
( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( x  e.  H  /\  f  ~~>v  x ) ) )
1312eximdv 1772 . . . . . . . . . 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 2762 . . . . . . . . 9  |-  ( E. x  e.  H  f 
~~>v  x  <->  E. x ( x  e.  H  /\  f  ~~>v  x ) )
1614, 15syl6ibr 235 . . . . . . . 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 441 . . . . . 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 1769 . . . . . . . 8  |-  F/ x  f  e.  Cauchy
20 nfv 1769 . . . . . . . . 9  |-  F/ x  f : NN --> H
21 nfre1 2846 . . . . . . . . 9  |-  F/ x E. x  e.  H  f  ~~>v  x
2220, 21nfim 2023 . . . . . . . 8  |-  F/ x
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )
2319, 22nfim 2023 . . . . . . 7  |-  F/ x
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
24 bi2.04 368 . . . . . . . . 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 26970 . . . . . . . . . . . 12  |-  ( f 
~~>v  x  ->  f  e.  Cauchy )
2625imim1i 59 . . . . . . . . . . 11  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  E. x  e.  H  f  ~~>v  x ) )
27 rexex 2843 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x  f  ~~>v  x )
28 hlimeui 26974 . . . . . . . . . . . . 13  |-  ( E. x  f  ~~>v  x  <->  E! x  f  ~~>v  x )
2927, 28sylib 201 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E! x  f  ~~>v  x )
30 exancom 1730 . . . . . . . . . . . . 13  |-  ( E. x ( x  e.  H  /\  f  ~~>v  x )  <->  E. x ( f 
~~>v  x  /\  x  e.  H ) )
3115, 30sylbb 202 . . . . . . . . . . . 12  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x
( f  ~~>v  x  /\  x  e.  H )
)
32 eupick 2385 . . . . . . . . . . . 12  |-  ( ( E! x  f  ~~>v  x  /\  E. x ( f  ~~>v  x  /\  x  e.  H ) )  -> 
( f  ~~>v  x  ->  x  e.  H )
)
3329, 31, 32syl2anc 673 . . . . . . . . . . 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 16 . . . . . . . . 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 200 . . . . . . . 8  |-  ( ( f  e.  Cauchy  ->  (
f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )  ->  ( f : NN --> H  ->  (
f  ~~>v  x  ->  x  e.  H ) ) )
3736impd 438 . . . . . . 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 1975 . . . . . 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 192 . . . . 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 1699 . . . 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 2761 . . . 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 260 . . 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 708 . 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 257 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 189    /\ wa 376   A.wal 1450   E.wex 1671    e. wcel 1904   E!weu 2319   A.wral 2756   E.wrex 2757   class class class wbr 4395   -->wf 5585   NNcn 10631   ~Hchil 26653   Cauchyccau 26660    ~~>v chli 26661   SHcsh 26662   CHcch 26663
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  ax-hilex 26733  ax-hfvadd 26734  ax-hvcom 26735  ax-hvass 26736  ax-hv0cl 26737  ax-hvaddid 26738  ax-hfvmul 26739  ax-hvmulid 26740  ax-hvmulass 26741  ax-hvdistr1 26742  ax-hvdistr2 26743  ax-hvmul0 26744  ax-hfi 26813  ax-his1 26816  ax-his2 26817  ax-his3 26818  ax-his4 26819  ax-hcompl 26936
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-4 10692  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-icc 11667  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-haus 20408  df-cau 22304  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-hnorm 26702  df-hvsub 26705  df-hlim 26706  df-hcau 26707  df-ch 26955
This theorem is referenced by:  chcompl  26976  occl  27038
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