Theorem sheli 31424

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 31423	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3933	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2143   ℋchba 31129   S_ℋ csh 31138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-hilex 31209

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-sh 31417

This theorem is referenced by:  norm1exi  31460  hhssabloi  31472  hhssnv  31474  shscli  31527  shunssi  31578  shmodsi  31599  omlsii  31613  5oalem1  31864  5oalem2  31865  5oalem3  31866  5oalem5  31868  imaelshi  32268  pjimai  32386  shatomici  32568  shatomistici  32571  cdjreui  32642  cdj1i  32643  cdj3lem1  32644  cdj3lem2b  32647  cdj3lem3  32648  cdj3lem3b  32650  cdj3i  32651