Theorem sheli 31274

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 31273	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3918	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 30979   S_ℋ csh 30988

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 5231  ax-hilex 31059

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 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-sh 31267

This theorem is referenced by:  norm1exi  31310  hhssabloi  31322  hhssnv  31324  shscli  31377  shunssi  31428  shmodsi  31449  omlsii  31463  5oalem1  31714  5oalem2  31715  5oalem3  31716  5oalem5  31718  imaelshi  32118  pjimai  32236  shatomici  32418  shatomistici  32421  cdjreui  32492  cdj1i  32493  cdj3lem1  32494  cdj3lem2b  32497  cdj3lem3  32498  cdj3lem3b  32500  cdj3i  32501