Theorem sheli 31194

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 31193	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3925	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2111   ℋchba 30899   S_ℋ csh 30908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-hilex 30979

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-sh 31187

This theorem is referenced by:  norm1exi  31230  hhssabloi  31242  hhssnv  31244  shscli  31297  shunssi  31348  shmodsi  31369  omlsii  31383  5oalem1  31634  5oalem2  31635  5oalem3  31636  5oalem5  31638  imaelshi  32038  pjimai  32156  shatomici  32338  shatomistici  32341  cdjreui  32412  cdj1i  32413  cdj3lem1  32414  cdj3lem2b  32417  cdj3lem3  32418  cdj3lem3b  32420  cdj3i  32421