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Theorem sheli 27959
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 𝐻S
Assertion
Ref Expression
sheli (𝐴𝐻𝐴 ∈ ℋ)

Proof of Theorem sheli
StepHypRef Expression
1 shssi.1 . . 3 𝐻S
21shssii 27958 . 2 𝐻 ⊆ ℋ
32sseli 3584 1 (𝐴𝐻𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  chil 27664   S csh 27673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-hilex 27744
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-sh 27952
This theorem is referenced by:  norm1exi  27995  hhssabloi  28007  hhssnv  28009  shscli  28064  shunssi  28115  shmodsi  28136  omlsii  28150  5oalem1  28401  5oalem2  28402  5oalem3  28403  5oalem5  28405  imaelshi  28805  pjimai  28923  shatomici  29105  shatomistici  29108  cdjreui  29179  cdj1i  29180  cdj3lem1  29181  cdj3lem2b  29184  cdj3lem3  29185  cdj3lem3b  29187  cdj3i  29188
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