Theorem sheli 31195

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 31194	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3954	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2108   ℋchba 30900   S_ℋ csh 30909

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-hilex 30980

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-sh 31188

This theorem is referenced by:  norm1exi  31231  hhssabloi  31243  hhssnv  31245  shscli  31298  shunssi  31349  shmodsi  31370  omlsii  31384  5oalem1  31635  5oalem2  31636  5oalem3  31637  5oalem5  31639  imaelshi  32039  pjimai  32157  shatomici  32339  shatomistici  32342  cdjreui  32413  cdj1i  32414  cdj3lem1  32415  cdj3lem2b  32418  cdj3lem3  32419  cdj3lem3b  32421  cdj3i  32422