Theorem chel 31196

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 15-Dec-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
chel	⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem chel

Step	Hyp	Ref	Expression
1		chss 31195	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3965	1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2107   ℋchba 30885   C_ℋ cch 30895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5278  ax-hilex 30965

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fv 6550  df-ov 7417  df-sh 31173  df-ch 31187

This theorem is referenced by:  pjhtheu2  31382  pjspansn  31543  pjid  31661  atom1d  32319  sumdmdii  32381