HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chne0i Unicode version

Theorem chne0i 22032
Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
ch0le.1  |-  A  e. 
CH
Assertion
Ref Expression
chne0i  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Distinct variable group:    x, A

Proof of Theorem chne0i
StepHypRef Expression
1 ch0le.1 . . 3  |-  A  e. 
CH
21chshii 21807 . 2  |-  A  e.  SH
32shne0i 22027 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684    =/= wne 2446   E.wrex 2544   0hc0v 21504   CHcch 21509   0Hc0h 21515
This theorem is referenced by:  chne0  22073  hne0  22126  h1datomi  22160  riesz3i  22642  pjnmopi  22728
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-hilex 21579  ax-hv0cl 21583
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-sh 21786  df-ch 21801  df-ch0 21832
  Copyright terms: Public domain W3C validator