Theorem shel 5120

Description: A member of a subspace of a Hilbert space is a vector.

Hypothesis

Ref	Expression
shssi.1	⊢ H ∈ Sℋ

Assertion

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

Proof of Theorem shel

Step	Hyp	Ref	Expression
1		shssi.1	. . 3 ⊢ H ∈ Sℋ
2	1	shssi 5119	. 2 ⊢ H ⊆ ℋ
3	2	sseli 1504	1 ⊢ (A ∈ H → A ∈ ℋ )

Syntax hints:   → wi 2   ∈ wcel 1092   ℋ chil 4958   S_ℋ csh 4967

This theorem is referenced by:  chocuni 5179  omlsi 5250  pjoml 5271  shscl 5282  shunss 5338  shmods 5363  5oalem1 5544  5oalem2 5545  5oalem3 5546  5oalem5 5548  shatomic 5753

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114

metamath.org