Theorem shelt 5118

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

Assertion

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

Proof of Theorem shelt

Step	Hyp	Ref	Expression
1		shss 5117	. . 3 ⊢ (H ∈ Sℋ → H ⊆ ℋ )
2	1	sseld 1506	. 2 ⊢ (H ∈ Sℋ → (A ∈ H → A ∈ ℋ ))
3	2	imp 277	1 ⊢ ((H ∈ Sℋ ∧ A ∈ H) → A ∈ ℋ )

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092   ℋ chil 4958   S_ℋ csh 4967

This theorem is referenced by:  shscomt 5284  shsel1t 5286  elspanclt 5306  sh1dle 5748

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

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