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Theorem isch2 26258
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 26257 . 2  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  (  ~~>v  "
( H  ^m  NN ) )  C_  H
) )
2 alcom 1853 . . . . 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 1768 . . . . . . . 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 3037 . . . . . . . . . 10  |-  x  e. 
_V
54elima2 5255 . . . . . . . . 9  |-  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  <->  E. f ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x ) )
65imbi1i 323 . . . . . . . 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 252 . . . . . . 7  |-  ( A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H
) )
87albii 1648 . . . . . 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 3406 . . . . . 6  |-  ( ( 
~~>v  " ( H  ^m  NN ) )  C_  H  <->  A. x ( x  e.  (  ~~>v  " ( H  ^m  NN ) )  ->  x  e.  H ) )
108, 9bitr4i 252 . . . . 5  |-  ( A. x A. f ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
112, 10bitri 249 . . . 4  |-  ( A. f A. x ( ( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  ->  x  e.  H )  <->  ( 
~~>v  " ( H  ^m  NN ) )  C_  H
)
12 nnex 10458 . . . . . . . 8  |-  NN  e.  _V
13 elmapg 7351 . . . . . . . 8  |-  ( ( H  e.  SH  /\  NN  e.  _V )  -> 
( f  e.  ( H  ^m  NN )  <-> 
f : NN --> H ) )
1412, 13mpan2 669 . . . . . . 7  |-  ( H  e.  SH  ->  (
f  e.  ( H  ^m  NN )  <->  f : NN
--> H ) )
1514anbi1d 702 . . . . . 6  |-  ( H  e.  SH  ->  (
( f  e.  ( H  ^m  NN )  /\  f  ~~>v  x )  <-> 
( f : NN --> H  /\  f  ~~>v  x ) ) )
1615imbi1d 315 . . . . 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 1723 . . . 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 259 . . 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 635 . 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 249 1  |-  ( H  e.  CH  <->  ( H  e.  SH  /\  A. f A. x ( ( f : NN --> H  /\  f  ~~>v  x )  ->  x  e.  H )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1397   E.wex 1620    e. wcel 1826   _Vcvv 3034    C_ wss 3389   class class class wbr 4367   "cima 4916   -->wf 5492  (class class class)co 6196    ^m cmap 7338   NNcn 10452    ~~>v chli 25961   SHcsh 25962   CHcch 25963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-i2m1 9471  ax-1ne0 9472  ax-rrecex 9475  ax-cnre 9476
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-map 7340  df-nn 10453  df-ch 26256
This theorem is referenced by:  chlimi  26269  isch3  26276  helch  26278  hsn0elch  26283  chintcli  26366  chscl  26676  nlelchi  27096  hmopidmchi  27186
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