Theorem sheli 28596

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 28595	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3794	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2157   ℋchba 28301   S_ℋ csh 28310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-hilex 28381

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-sh 28589

This theorem is referenced by:  norm1exi  28632  hhssabloi  28644  hhssnv  28646  shscli  28701  shunssi  28752  shmodsi  28773  omlsii  28787  5oalem1  29038  5oalem2  29039  5oalem3  29040  5oalem5  29042  imaelshi  29442  pjimai  29560  shatomici  29742  shatomistici  29745  cdjreui  29816  cdj1i  29817  cdj3lem1  29818  cdj3lem2b  29821  cdj3lem3  29822  cdj3lem3b  29824  cdj3i  29825