Theorem shelii 31576

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 31574	. 2 ⊢ 𝐻 ⊆ ℋ
3		sheli.1	. 2 ⊢ 𝐴 ∈ 𝐻
4	2, 3	sselii 3934	1 ⊢ 𝐴 ∈ ℋ

This proof depends on syntax axioms:   ∈ wcel 2143   ℋchba 31280   S_ℋ csh 31289

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-hilex 31360

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-sh 31568

This theorem is used by:  omlsilem  31763