Theorem sheli 31285

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 31284	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3918	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 30990   S_ℋ csh 30999

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 5232  ax-hilex 31070

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-sh 31278

This theorem is referenced by:  norm1exi  31321  hhssabloi  31333  hhssnv  31335  shscli  31388  shunssi  31439  shmodsi  31460  omlsii  31474  5oalem1  31725  5oalem2  31726  5oalem3  31727  5oalem5  31729  imaelshi  32129  pjimai  32247  shatomici  32429  shatomistici  32432  cdjreui  32503  cdj1i  32504  cdj3lem1  32505  cdj3lem2b  32508  cdj3lem3  32509  cdj3lem3b  32511  cdj3i  32512