Theorem shel 31818

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 31817	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3908	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   ℋchba 31526   S_ℋ csh 31535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-hilex 31606

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-sh 31814

This theorem is used by:  shuni  31907  shsel3  31922  shscom  31926  shsel1  31928  elspancl  31944  pjpjpre  32026  spansnss  32178  sh1dle  32958