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Theorem isch2 22714
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 22713 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
2 alcom 1752 . . . . 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 1914 . . . . . . . 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 2951 . . . . . . . . . 10  |-  x  e. 
_V
54elima2 5200 . . . . . . . . 9  |-  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  <->  E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x ) )
65imbi1i 316 . . . . . . . 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 244 . . . . . . 7  |-  ( A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H
) )
87albii 1575 . . . . . 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 3329 . . . . . 6  |-  ( ( 
~~>v  " ( H  ^m  NN ) )  C_  H  <->  A. x ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H ) )
108, 9bitr4i 244 . . . . 5  |-  ( A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
112, 10bitri 241 . . . 4  |-  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
12 nnex 9995 . . . . . . . 8  |-  NN  e.  _V
13 elmapg 7022 . . . . . . . 8  |-  ( ( H  e.  SH  /\  NN  e.  _V )  -> 
( f  e.  ( H  ^m  NN )  <-> 
f : NN --> H ) )
1412, 13mpan2 653 . . . . . . 7  |-  ( H  e.  SH  ->  (
f  e.  ( H  ^m  NN )  <->  f : NN
--> H ) )
1514anbi1d 686 . . . . . 6  |-  ( H  e.  SH  ->  (
( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  <-> 
( f : NN --> H  /\  f  ~~>v  x ) ) )
1615imbi1d 309 . . . . 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 1637 . . . 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 251 . . 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 619 . 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 241 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 177    /\ wa 359   A.wal 1549   E.wex 1550    e. wcel 1725   _Vcvv 2948    C_ wss 3312   class class class wbr 4204   "cima 4872   -->wf 5441  (class class class)co 6072    ^m cmap 7009   NNcn 9989    ~~>v chli 22418   SHcsh 22419   CHcch 22420
This theorem is referenced by:  chlimi  22725  isch3  22732  helch  22734  hsn0elch  22738  chintcli  22821  chscl  23131  nlelchi  23552  hmopidmchi  23642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-i2m1 9047  ax-1ne0 9048  ax-rrecex 9051  ax-cnre 9052
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-recs 6624  df-rdg 6659  df-map 7011  df-nn 9990  df-ch 22712
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