Theorem chel 31765

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 31764	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3930	1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ℋchba 31454   C_ℋ cch 31464

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-hilex 31534

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fv 6535  df-ov 7411  df-sh 31742  df-ch 31756

This theorem is used by:  pjhtheu2  31951  pjspansn  32112  pjid  32230  atom1d  32888  sumdmdii  32950