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

Theorem chne0i 22048
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 21823 . 2  |-  A  e.  SH
32shne0i 22043 1  |-  ( A  =/=  0H  <->  E. x  e.  A  x  =/=  0h )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1696    =/= wne 2459   E.wrex 2557   0hc0v 21520   CHcch 21525   0Hc0h 21531
This theorem is referenced by:  chne0  22089  hne0  22142  h1datomi  22176  riesz3i  22658  pjnmopi  22744
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-hilex 21595  ax-hv0cl 21599
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-sh 21802  df-ch 21817  df-ch0 21848
  Copyright terms: Public domain W3C validator