Theorem chel 29724

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 29723	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3930	1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2105   ℋchba 29413   C_ℋ cch 29423

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5237  ax-hilex 29493

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3442  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-br 5087  df-opab 5149  df-xp 5613  df-cnv 5615  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fv 6473  df-ov 7319  df-sh 29701  df-ch 29715

This theorem is referenced by:  pjhtheu2  29910  pjspansn  30071  pjid  30189  atom1d  30847  sumdmdii  30909