Theorem sheli 31846

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

Hypothesis

Ref	Expression
shssi.1	⊢ 𝐻 ∈ Sℋ

Assertion

Ref	Expression
sheli	⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Proof of Theorem sheli

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 ⊢ 𝐻 ∈ Sℋ
2	1	shssii 31845	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3909	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ℋchba 31551   S_ℋ csh 31560

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-hilex 31631

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-sh 31839

This theorem is used by:  norm1exi  31882  hhssabloi  31894  hhssnv  31896  shscli  31949  shunssi  32000  shmodsi  32021  omlsii  32035  5oalem1  32286  5oalem2  32287  5oalem3  32288  5oalem5  32290  imaelshi  32690  pjimai  32808  shatomici  32990  shatomistici  32993  cdjreui  33064  cdj1i  33065  cdj3lem1  33066  cdj3lem2b  33069  cdj3lem3  33070  cdj3lem3b  33072  cdj3i  33073