Theorem sheli 31270

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 31269	. 2 ⊢ 𝐻 ⊆ ℋ
3	2	sseli 3928	1 ⊢ (𝐴 ∈ 𝐻 → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∈ wcel 2114   ℋchba 30975   S_ℋ csh 30984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-hilex 31055

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-sh 31263

This theorem is referenced by:  norm1exi  31306  hhssabloi  31318  hhssnv  31320  shscli  31373  shunssi  31424  shmodsi  31445  omlsii  31459  5oalem1  31710  5oalem2  31711  5oalem3  31712  5oalem5  31714  imaelshi  32114  pjimai  32232  shatomici  32414  shatomistici  32417  cdjreui  32488  cdj1i  32489  cdj3lem1  32490  cdj3lem2b  32493  cdj3lem3  32494  cdj3lem3b  32496  cdj3i  32497