Theorem shelii 31148

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	⊢ 𝐻 ∈ Sℋ
sheli.1	⊢ 𝐴 ∈ 𝐻

Assertion

Ref	Expression
shelii	⊢ 𝐴 ∈ ℋ

Proof of Theorem shelii

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 ⊢ 𝐻 ∈ Sℋ
2	1	shssii 31146	. 2 ⊢ 𝐻 ⊆ ℋ
3		sheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3976	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2099   ℋchba 30852   S_ℋ csh 30861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-hilex 30932

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-sh 31140

This theorem is referenced by:  omlsilem  31335