Theorem shel 31297

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 14-Dec-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
shel	⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Proof of Theorem shel

Step	Hyp	Ref	Expression
1		shss 31296	. 2 ⊢ (𝐻 ∈ Sℋ → 𝐻 ⊆ ℋ)
2	1	sselda 3922	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻) → 𝐴 ∈ ℋ)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ℋchba 31005   S_ℋ csh 31014

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 2709  ax-sep 5231  ax-hilex 31085

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-sh 31293

This theorem is referenced by:  shuni  31386  shsel3  31401  shscom  31405  shsel1  31407  elspancl  31423  pjpjpre  31505  spansnss  31657  sh1dle  32437