Theorem sheli 31143

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 31142	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3942	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30848   S_ℋ csh 30857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-hilex 30928

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-sh 31136

This theorem is referenced by:  norm1exi  31179  hhssabloi  31191  hhssnv  31193  shscli  31246  shunssi  31297  shmodsi  31318  omlsii  31332  5oalem1  31583  5oalem2  31584  5oalem3  31585  5oalem5  31587  imaelshi  31987  pjimai  32105  shatomici  32287  shatomistici  32290  cdjreui  32361  cdj1i  32362  cdj3lem1  32363  cdj3lem2b  32366  cdj3lem3  32367  cdj3lem3b  32369  cdj3i  32370