Theorem sh0

Description: The zero vector belongs to any subspace of a Hilbert space. (Contributed by NM, 11-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	sh0	⊢ ( 𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		issh	⊢ ( 𝐻 ∈ Sℋ ↔ ( ( 𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻 ) ∧ ( ( +ℎ “ ( 𝐻 × 𝐻 ) ) ⊆ 𝐻 ∧ ( ·ℎ “ ( ℂ × 𝐻 ) ) ⊆ 𝐻 ) ) )
2	1	simplbi	⊢ ( 𝐻 ∈ Sℋ → ( 𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻 ) )
3	2	simprd	⊢ ( 𝐻 ∈ Sℋ → 0ℎ ∈ 𝐻 )