Theorem shelii 28995

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 28993	. 2 ⊢ 𝐻 ⊆ ℋ
3		sheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3967	1 ⊢ 𝐴 ∈ ℋ

Syntax hints:   ∈ wcel 2113   ℋchba 28699   S_ℋ csh 28708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-hilex 28779

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-sh 28987

This theorem is referenced by:  omlsilem  29182