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Theorem shelii 28627
 Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
shssi.1 𝐻S
sheli.1 𝐴𝐻
Assertion
Ref Expression
shelii 𝐴 ∈ ℋ

Proof of Theorem shelii
StepHypRef Expression
1 shssi.1 . . 3 𝐻S
21shssii 28625 . 2 𝐻 ⊆ ℋ
3 sheli.1 . 2 𝐴𝐻
42, 3sselii 3824 1 𝐴 ∈ ℋ
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2166   ℋchba 28331   Sℋ csh 28340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-hilex 28411 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-xp 5348  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-sh 28619 This theorem is referenced by:  omlsilem  28816
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