Theorem sheli 29477

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 29476	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3913	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2108   ℋchba 29182   S_ℋ csh 29191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-hilex 29262

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-sh 29470

This theorem is referenced by:  norm1exi  29513  hhssabloi  29525  hhssnv  29527  shscli  29580  shunssi  29631  shmodsi  29652  omlsii  29666  5oalem1  29917  5oalem2  29918  5oalem3  29919  5oalem5  29921  imaelshi  30321  pjimai  30439  shatomici  30621  shatomistici  30624  cdjreui  30695  cdj1i  30696  cdj3lem1  30697  cdj3lem2b  30700  cdj3lem3  30701  cdj3lem3b  30703  cdj3i  30704