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Theorem cheli 26761
Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypothesis
Ref Expression
chssi.1  |-  H  e. 
CH
Assertion
Ref Expression
cheli  |-  ( A  e.  H  ->  A  e.  ~H )

Proof of Theorem cheli
StepHypRef Expression
1 chssi.1 . . 3  |-  H  e. 
CH
21chssii 26760 . 2  |-  H  C_  ~H
32sseli 3457 1  |-  ( A  e.  H  ->  A  e.  ~H )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   ~Hchil 26448   CHcch 26458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-hilex 26528
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fv 5600  df-ov 6299  df-sh 26736  df-ch 26750
This theorem is referenced by:  pjhthlem1  26920  pjhthlem2  26921  h1de2ci  27085  spanunsni  27108  spansncvi  27181  3oalem1  27191  pjcompi  27201  pjocini  27227  pjjsi  27229  pjrni  27231  pjdsi  27241  pjds3i  27242  mayete3i  27257  riesz3i  27591  pjnmopi  27677  pjnormssi  27697  pjimai  27705  pjclem4a  27727  pjclem4  27728  pj3lem1  27735  pj3si  27736  strlem1  27779  strlem3  27782  strlem5  27784  hstrlem3  27790  hstrlem5  27792  sumdmdii  27944  sumdmdlem  27947  sumdmdlem2  27948
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