Theorem shel 31472

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 31471	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3939	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145   ℋchba 31180   S_ℋ csh 31189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-hilex 31260

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-sh 31468

This theorem is referenced by:  shuni  31561  shsel3  31576  shscom  31580  shsel1  31582  elspancl  31598  pjpjpre  31680  spansnss  31832  sh1dle  32612