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

Theorem sheli 22557
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
shssi.1  |-  H  e.  SH
Assertion
Ref Expression
sheli  |-  ( A  e.  H  ->  A  e.  ~H )

Proof of Theorem sheli
StepHypRef Expression
1 shssi.1 . . 3  |-  H  e.  SH
21shssii 22556 . 2  |-  H  C_  ~H
32sseli 3280 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   ~Hchil 22263   SHcsh 22272
This theorem is referenced by:  norm1exi  22593  hhssabloi  22603  hhssnv  22605  shscli  22660  shunssi  22711  shmodsi  22732  omlsii  22746  5oalem1  22997  5oalem2  22998  5oalem3  22999  5oalem5  23001  imaelshi  23402  pjimai  23520  shatomici  23702  shatomistici  23705  cdjreui  23776  cdj1i  23777  cdj3lem1  23778  cdj3lem2b  23781  cdj3lem3  23782  cdj3lem3b  23784  cdj3i  23785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-hilex 22343
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-cnv 4819  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-sh 22550
  Copyright terms: Public domain W3C validator