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Theorem shel 28408
Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)
Assertion
Ref Expression
shel ((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem shel
StepHypRef Expression
1 shss 28407 . 2 (𝐻S𝐻 ⊆ ℋ)
21sselda 3752 1 ((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  chil 28116   S csh 28125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-hilex 28196
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-sh 28404
This theorem is referenced by:  shuni  28499  shsel3  28514  shscom  28518  shsel1  28520  elspancl  28536  pjpjpre  28618  spansnss  28770  sh1dle  29550
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