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Theorem isch2 21819
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 21818 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
2 alcom 1723 . . . . 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 1844 . . . . . . . 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 2804 . . . . . . . . . 10  |-  x  e. 
_V
54elima2 5034 . . . . . . . . 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 1556 . . . . . 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 3182 . . . . . 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 9768 . . . . . . . 8  |-  NN  e.  _V
13 elmapg 6801 . . . . . . . 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 1617 . . . 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 1530   E.wex 1531    e. wcel 1696   _Vcvv 2801    C_ wss 3165   class class class wbr 4039   "cima 4708   -->wf 5267  (class class class)co 5874    ^m cmap 6788   NNcn 9762    ~~>v chli 21523   SHcsh 21524   CHcch 21525
This theorem is referenced by:  chlimi  21830  isch3  21837  helch  21839  hsn0elch  21843  chintcli  21926  chscl  22236  nlelchi  22657  hmopidmchi  22747
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-map 6790  df-nn 9763  df-ch 21817
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