Theorem sheli 31532

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 31531	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3932	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2141   ℋchba 31237   S_ℋ csh 31246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-hilex 31317

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-sh 31525

This theorem is referenced by:  norm1exi  31568  hhssabloi  31580  hhssnv  31582  shscli  31635  shunssi  31686  shmodsi  31707  omlsii  31721  5oalem1  31972  5oalem2  31973  5oalem3  31974  5oalem5  31976  imaelshi  32376  pjimai  32494  shatomici  32676  shatomistici  32679  cdjreui  32750  cdj1i  32751  cdj3lem1  32752  cdj3lem2b  32755  cdj3lem3  32756  cdj3lem3b  32758  cdj3i  32759