Theorem sheli 31294

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 31293	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3930	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 30999   S_ℋ csh 31008

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-hilex 31079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-sh 31287

This theorem is referenced by:  norm1exi  31330  hhssabloi  31342  hhssnv  31344  shscli  31397  shunssi  31448  shmodsi  31469  omlsii  31483  5oalem1  31734  5oalem2  31735  5oalem3  31736  5oalem5  31738  imaelshi  32138  pjimai  32256  shatomici  32438  shatomistici  32441  cdjreui  32512  cdj1i  32513  cdj3lem1  32514  cdj3lem2b  32517  cdj3lem3  32518  cdj3lem3b  32520  cdj3i  32521