Theorem sheli 31242

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 31241	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3990	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2105   ℋchba 30947   S_ℋ csh 30956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-hilex 31027

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-sh 31235

This theorem is referenced by:  norm1exi  31278  hhssabloi  31290  hhssnv  31292  shscli  31345  shunssi  31396  shmodsi  31417  omlsii  31431  5oalem1  31682  5oalem2  31683  5oalem3  31684  5oalem5  31686  imaelshi  32086  pjimai  32204  shatomici  32386  shatomistici  32389  cdjreui  32460  cdj1i  32461  cdj3lem1  32462  cdj3lem2b  32465  cdj3lem3  32466  cdj3lem3b  32468  cdj3i  32469