Theorem shelii 31680

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

Hypotheses

Ref	Expression
shssi.1	⊢ 𝐻 ∈ Sℋ
sheli.1	⊢ 𝐴 ∈ 𝐻

Assertion

Ref	Expression
shelii	⊢ 𝐴 ∈ ℋ

Proof of Theorem shelii

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 ⊢ 𝐻 ∈ Sℋ
2	1	shssii 31678	. 2 ⊢ 𝐻 ⊆ ℋ
3		sheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3931	1 ⊢ 𝐴 ∈ ℋ

This proof depends on syntax axioms:   ∈ wcel 2145   ℋchba 31384   S_ℋ csh 31393

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 2734  ax-sep 5255  ax-hilex 31464

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 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-sh 31672

This theorem is used by:  omlsilem  31867