Theorem sheli 31696

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 31695	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3930	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ℋchba 31401   S_ℋ csh 31410

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 31481

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 31689

This theorem is used by:  norm1exi  31732  hhssabloi  31744  hhssnv  31746  shscli  31799  shunssi  31850  shmodsi  31871  omlsii  31885  5oalem1  32136  5oalem2  32137  5oalem3  32138  5oalem5  32140  imaelshi  32540  pjimai  32658  shatomici  32840  shatomistici  32843  cdjreui  32914  cdj1i  32915  cdj3lem1  32916  cdj3lem2b  32919  cdj3lem3  32920  cdj3lem3b  32922  cdj3i  32923