Theorem chelii 31817

Description: A member of a closed subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
chssi.1	⊢ 𝐻 ∈ Cℋ
cheli.1	⊢ 𝐴 ∈ 𝐻

Assertion

Ref	Expression
chelii	⊢ 𝐴 ∈ ℋ

Proof of Theorem chelii

Step	Hyp	Ref	Expression
1		chssi.1	. . 3 ⊢ 𝐻 ∈ Cℋ
2	1	chssii 31815	. 2 ⊢ 𝐻 ⊆ ℋ
3		cheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3928	1 ⊢ 𝐴 ∈ ℋ

This proof depends on syntax axioms:   ∈ wcel 2145   ℋchba 31503   C_ℋ cch 31513

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 2733  ax-sep 5249  ax-hilex 31583

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fv 6539  df-ov 7415  df-sh 31791  df-ch 31805

This theorem is used by: (None)