Theorem sheli 30467

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 30466	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3979	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2107   ℋchba 30172   S_ℋ csh 30181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-hilex 30252

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-sh 30460

This theorem is referenced by:  norm1exi  30503  hhssabloi  30515  hhssnv  30517  shscli  30570  shunssi  30621  shmodsi  30642  omlsii  30656  5oalem1  30907  5oalem2  30908  5oalem3  30909  5oalem5  30911  imaelshi  31311  pjimai  31429  shatomici  31611  shatomistici  31614  cdjreui  31685  cdj1i  31686  cdj3lem1  31687  cdj3lem2b  31690  cdj3lem3  31691  cdj3lem3b  31693  cdj3i  31694