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Theorem isch3 26559
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 26541 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
2 ax-hcompl 26519 . . . . . . . . . 10  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
3 rexex 2860 . . . . . . . . . 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 1704 . . . . . . . . 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 472 . . . . . . . 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 427 . . . . . . . . . . . . . 14  |-  ( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  (
f : NN --> H  /\  f  ~~>v  x ) )  ->  x  e.  H
)
98an12s 802 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  x  e.  H )
10 simprr 758 . . . . . . . . . . . . 13  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  f  ~~>v  x )
119, 10jca 530 . . . . . . . . . . . 12  |-  ( ( f : NN --> H  /\  ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x ) )  ->  ( x  e.  H  /\  f  ~~>v  x ) )
1211ex 432 . . . . . . . . . . 11  |-  ( f : NN --> H  -> 
( ( ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )  /\  f  ~~>v  x )  ->  ( x  e.  H  /\  f  ~~>v  x ) ) )
1312eximdv 1731 . . . . . . . . . 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 2759 . . . . . . . . 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 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 432 . . . . . 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 1728 . . . . . . . 8  |-  F/ x  f  e.  Cauchy
20 nfv 1728 . . . . . . . . 9  |-  F/ x  f : NN --> H
21 nfre1 2864 . . . . . . . . 9  |-  F/ x E. x  e.  H  f  ~~>v  x
2220, 21nfim 1948 . . . . . . . 8  |-  F/ x
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x )
2319, 22nfim 1948 . . . . . . 7  |-  F/ x
( f  e.  Cauchy  -> 
( f : NN --> H  ->  E. x  e.  H  f  ~~>v  x ) )
24 bi2.04 359 . . . . . . . . 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 26554 . . . . . . . . . . . 12  |-  ( f 
~~>v  x  ->  f  e.  Cauchy )
2625imim1i 57 . . . . . . . . . . 11  |-  ( ( f  e.  Cauchy  ->  E. x  e.  H  f  ~~>v  x )  ->  ( f  ~~>v  x  ->  E. x  e.  H  f  ~~>v  x ) )
27 rexex 2860 . . . . . . . . . . . . 13  |-  ( E. x  e.  H  f 
~~>v  x  ->  E. x  f  ~~>v  x )
28 hlimeui 26558 . . . . . . . . . . . . 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 1692 . . . . . . . . . . . . 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 2309 . . . . . . . . . . . 12  |-  ( ( E! x  f  ~~>v  x  /\  E. x ( f  ~~>v  x  /\  x  e.  H ) )  -> 
( f  ~~>v  x  ->  x  e.  H )
)
3329, 31, 32syl2anc 659 . . . . . . . . . . 11  |-  ( E. x  e.  H  f 
~~>v  x  ->  ( f  ~~>v  x  ->  x  e.  H ) )
3426, 33syli 35 . . . . . . . . . 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 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 429 . . . . . . 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 1901 . . . . . 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 1661 . . . 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 2758 . . . 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 692 . 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 367   A.wal 1403   E.wex 1633    e. wcel 1842   E!weu 2238   A.wral 2753   E.wrex 2754   class class class wbr 4394   -->wf 5564   NNcn 10575   ~Hchil 26236   Cauchyccau 26243    ~~>v chli 26244   SHcsh 26245   CHcch 26246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599  ax-addf 9600  ax-mulf 9601  ax-hilex 26316  ax-hfvadd 26317  ax-hvcom 26318  ax-hvass 26319  ax-hv0cl 26320  ax-hvaddid 26321  ax-hfvmul 26322  ax-hvmulid 26323  ax-hvmulass 26324  ax-hvdistr1 26325  ax-hvdistr2 26326  ax-hvmul0 26327  ax-hfi 26396  ax-his1 26399  ax-his2 26400  ax-his3 26401  ax-his4 26402  ax-hcompl 26519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-sup 7934  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-n0 10836  df-z 10905  df-uz 11127  df-q 11227  df-rp 11265  df-xneg 11370  df-xadd 11371  df-xmul 11372  df-icc 11588  df-seq 12150  df-exp 12209  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-topgen 15056  df-psmet 18729  df-xmet 18730  df-met 18731  df-bl 18732  df-mopn 18733  df-top 19689  df-bases 19691  df-topon 19692  df-lm 20021  df-haus 20107  df-cau 21985  df-grpo 25593  df-gid 25594  df-ginv 25595  df-gdiv 25596  df-ablo 25684  df-vc 25839  df-nv 25885  df-va 25888  df-ba 25889  df-sm 25890  df-0v 25891  df-vs 25892  df-nmcv 25893  df-ims 25894  df-hnorm 26285  df-hvsub 26288  df-hlim 26289  df-hcau 26290  df-ch 26539
This theorem is referenced by:  chcompl  26560  occl  26622
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