Theorem sheli 28997

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
shssi.1	⊢ 𝐻 ∈ Sℋ

Assertion

Ref	Expression
sheli	⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Proof of Theorem sheli

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 ⊢ 𝐻 ∈ Sℋ
2	1	shssii 28996	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3911	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2111   ℋchba 28702   S_ℋ csh 28711

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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-hilex 28782

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 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-sh 28990

This theorem is referenced by:  norm1exi  29033  hhssabloi  29045  hhssnv  29047  shscli  29100  shunssi  29151  shmodsi  29172  omlsii  29186  5oalem1  29437  5oalem2  29438  5oalem3  29439  5oalem5  29441  imaelshi  29841  pjimai  29959  shatomici  30141  shatomistici  30144  cdjreui  30215  cdj1i  30216  cdj3lem1  30217  cdj3lem2b  30220  cdj3lem3  30221  cdj3lem3b  30223  cdj3i  30224