Theorem sheli 31289

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 31288	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3929	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2113   ℋchba 30994   S_ℋ csh 31003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-hilex 31074

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-sh 31282

This theorem is referenced by:  norm1exi  31325  hhssabloi  31337  hhssnv  31339  shscli  31392  shunssi  31443  shmodsi  31464  omlsii  31478  5oalem1  31729  5oalem2  31730  5oalem3  31731  5oalem5  31733  imaelshi  32133  pjimai  32251  shatomici  32433  shatomistici  32436  cdjreui  32507  cdj1i  32508  cdj3lem1  32509  cdj3lem2b  32512  cdj3lem3  32513  cdj3lem3b  32515  cdj3i  32516