Theorem sheli 31703

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 31702	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3930	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ℋchba 31408   S_ℋ csh 31417

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 31488

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 31696

This theorem is used by:  norm1exi  31739  hhssabloi  31751  hhssnv  31753  shscli  31806  shunssi  31857  shmodsi  31878  omlsii  31892  5oalem1  32143  5oalem2  32144  5oalem3  32145  5oalem5  32147  imaelshi  32547  pjimai  32665  shatomici  32847  shatomistici  32850  cdjreui  32921  cdj1i  32922  cdj3lem1  32923  cdj3lem2b  32926  cdj3lem3  32927  cdj3lem3b  32929  cdj3i  32930