Theorem shel 30502

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
shel	⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem shel

Step	Hyp	Ref	Expression
1		shss 30501	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3982	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   ℋchba 30210   S_ℋ csh 30219

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-hilex 30290

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-sh 30498

This theorem is referenced by:  shuni  30591  shsel3  30606  shscom  30610  shsel1  30612  elspancl  30628  pjpjpre  30710  spansnss  30862  sh1dle  31642