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Metamath Proof Explorer

Theorem shelii

Description: A member of a subspace of a Hilbert space is a vector. (Contributed by NM, 6-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypotheses shssi.1 ⊢ H ∈ S ℋ
sheli.1 ⊢ A ∈ H
Assertion shelii ⊢ A ∈ ℋ

Proof

Step Hyp Ref Expression
1 shssi.1 ⊢ H ∈ S ℋ
2 sheli.1 ⊢ A ∈ H
3 1 shssii ⊢ H ⊆ ℋ
4 3 2 sselii ⊢ A ∈ ℋ