Theorem sheli 31162

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 31161	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3931	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30867   S_ℋ csh 30876

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 5235  ax-hilex 30947

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-sh 31155

This theorem is referenced by:  norm1exi  31198  hhssabloi  31210  hhssnv  31212  shscli  31265  shunssi  31316  shmodsi  31337  omlsii  31351  5oalem1  31602  5oalem2  31603  5oalem3  31604  5oalem5  31606  imaelshi  32006  pjimai  32124  shatomici  32306  shatomistici  32309  cdjreui  32380  cdj1i  32381  cdj3lem1  32382  cdj3lem2b  32385  cdj3lem3  32386  cdj3lem3b  32388  cdj3i  32389