HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  shss Unicode version

Theorem shss 21903
Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
shss  |-  ( H  e.  SH  ->  H  C_ 
~H )

Proof of Theorem shss
StepHypRef Expression
1 issh 21901 . . 3  |-  ( H  e.  SH  <->  ( ( H  C_  ~H  /\  0h  e.  H )  /\  (
(  +h  " ( H  X.  H ) ) 
C_  H  /\  (  .h  " ( CC  X.  H ) )  C_  H ) ) )
21simplbi 446 . 2  |-  ( H  e.  SH  ->  ( H  C_  ~H  /\  0h  e.  H ) )
32simpld 445 1  |-  ( H  e.  SH  ->  H  C_ 
~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710    C_ wss 3228    X. cxp 4769   "cima 4774   CCcc 8825   ~Hchil 21613    +h cva 21614    .h csm 21615   0hc0v 21618   SHcsh 21622
This theorem is referenced by:  shel  21904  shex  21905  shssii  21906  shsubcl  21914  chss  21923  shsspwh  21939  hhsssh  21960  shocel  21975  shocsh  21977  ocss  21978  shocss  21979  shocorth  21985  shococss  21987  shorth  21988  shoccl  21998  shsel  22007  shintcli  22022  spanid  22040  shjval  22044  shjcl  22049  shlej1  22053  shlub  22107  chscllem2  22331  chscllem4  22333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-hilex 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-sh 21900
  Copyright terms: Public domain W3C validator