Theorem sheli 31150

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 31149	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3945	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2109   ℋchba 30855   S_ℋ csh 30864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-hilex 30935

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-sh 31143

This theorem is referenced by:  norm1exi  31186  hhssabloi  31198  hhssnv  31200  shscli  31253  shunssi  31304  shmodsi  31325  omlsii  31339  5oalem1  31590  5oalem2  31591  5oalem3  31592  5oalem5  31594  imaelshi  31994  pjimai  32112  shatomici  32294  shatomistici  32297  cdjreui  32368  cdj1i  32369  cdj3lem1  32370  cdj3lem2b  32373  cdj3lem3  32374  cdj3lem3b  32376  cdj3i  32377