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Theorem isch2 21803
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 21802 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
2 alcom 1711 . . . . 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 1832 . . . . . . . 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 2791 . . . . . . . . . 10  |-  x  e. 
_V
54elima2 5018 . . . . . . . . 9  |-  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  <->  E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x ) )
65imbi1i 315 . . . . . . . 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 243 . . . . . . 7  |-  ( A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H
) )
87albii 1553 . . . . . 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 3169 . . . . . 6  |-  ( ( 
~~>v  " ( H  ^m  NN ) )  C_  H  <->  A. x ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H ) )
108, 9bitr4i 243 . . . . 5  |-  ( A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
112, 10bitri 240 . . . 4  |-  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
12 nnex 9752 . . . . . . . 8  |-  NN  e.  _V
13 elmapg 6785 . . . . . . . 8  |-  ( ( H  e.  SH  /\  NN  e.  _V )  -> 
( f  e.  ( H  ^m  NN )  <-> 
f : NN --> H ) )
1412, 13mpan2 652 . . . . . . 7  |-  ( H  e.  SH  ->  (
f  e.  ( H  ^m  NN )  <->  f : NN
--> H ) )
1514anbi1d 685 . . . . . 6  |-  ( H  e.  SH  ->  (
( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  <-> 
( f : NN --> H  /\  f  ~~>v  x ) ) )
1615imbi1d 308 . . . . 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 1613 . . . 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 250 . . 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 618 . 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 240 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 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    e. wcel 1684   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   "cima 4692   -->wf 5251  (class class class)co 5858    ^m cmap 6772   NNcn 9746    ~~>v chli 21507   SHcsh 21508   CHcch 21509
This theorem is referenced by:  chlimi  21814  isch3  21821  helch  21823  hsn0elch  21827  chintcli  21910  chscl  22220  nlelchi  22641  hmopidmchi  22731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-map 6774  df-nn 9747  df-ch 21801
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