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Metamath Proof Explorer

Theorem shss

Description: A subspace is a subset of Hilbert space. (Contributed by NM, 9-Oct-1999) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	shss	⊢ ( 𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ )

Proof

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