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Theorem isch2 21795
Description: Closed subspace  H of a Hilbert space. Definition of [Beran] p. 107. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
isch2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
Distinct variable group:    x, f, H

Proof of Theorem isch2
StepHypRef Expression
1 isch 21794 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
2 alcom 1715 . . . . 5  |-  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H
) )
3 19.23v 1843 . . . . . . . 8  |-  ( A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H
) )
4 vex 2792 . . . . . . . . . 10  |-  x  e. 
_V
54elima2 5017 . . . . . . . . 9  |-  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  <->  E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x ) )
65imbi1i 317 . . . . . . . 8  |-  ( ( x  e.  (  ~~>v  "
( H  ^m  NN ) )  ->  x  e.  H )  <->  ( E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H
) )
73, 6bitr4i 245 . . . . . . 7  |-  ( A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H
) )
87albii 1558 . . . . . 6  |-  ( A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  A. x ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H ) )
9 dfss2 3170 . . . . . 6  |-  ( ( 
~~>v  " ( H  ^m  NN ) )  C_  H  <->  A. x ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H ) )
108, 9bitr4i 245 . . . . 5  |-  ( A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
112, 10bitri 242 . . . 4  |-  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
12 nnex 9747 . . . . . . . 8  |-  NN  e.  _V
13 elmapg 6780 . . . . . . . 8  |-  ( ( H  e.  SH  /\  NN  e.  _V )  -> 
( f  e.  ( H  ^m  NN )  <-> 
f : NN --> H ) )
1412, 13mpan2 655 . . . . . . 7  |-  ( H  e.  SH  ->  (
f  e.  ( H  ^m  NN )  <->  f : NN
--> H ) )
1514anbi1d 688 . . . . . 6  |-  ( H  e.  SH  ->  (
( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  <-> 
( f : NN --> H  /\  f  ~~>v  x ) ) )
1615imbi1d 310 . . . . 5  |-  ( H  e.  SH  ->  (
( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H
)  <->  ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
17162albidv 1618 . . . 4  |-  ( H  e.  SH  ->  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H
)  <->  A. f A. x
( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H ) ) )
1811, 17syl5bbr 252 . . 3  |-  ( H  e.  SH  ->  (
(  ~~>v  " ( H  ^m  NN ) )  C_  H  <->  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H
) ) )
1918pm5.32i 621 . 2  |-  ( ( H  e.  SH  /\  (  ~~>v  " ( H  ^m  NN ) )  C_  H
)  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
201, 19bitri 242 1  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1533    e. wcel 1688   _Vcvv 2789    C_ wss 3153   class class class wbr 4024   "cima 4691   -->wf 5217  (class class class)co 5819    ^m cmap 6767   NNcn 9741    ~~>v chli 21499   SHcsh 21500   CHcch 21501
This theorem is referenced by:  chlimi  21806  isch3  21813  helch  21815  hsn0elch  21819  chintcli  21902  chscl  22212  nlelchi  22633  hmopidmchi  22723
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-i2m1 8800  ax-1ne0 8801  ax-rrecex 8804  ax-cnre 8805
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-map 6769  df-nn 9742  df-ch 21793
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