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Theorem shel 28992
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 28991 . 2 (𝐻S𝐻 ⊆ ℋ)
21sselda 3942 1 ((𝐻S𝐴𝐻) → 𝐴 ∈ ℋ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  chba 28700   S csh 28709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-hilex 28780
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rab 3139  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-sh 28988
This theorem is referenced by:  shuni  29081  shsel3  29096  shscom  29100  shsel1  29102  elspancl  29118  pjpjpre  29200  spansnss  29352  sh1dle  30132
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