Metamath Proof Explorer

Theorem shocsh

Description: The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of Beran p. 107. (Contributed by NM, 10-Oct-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	shocsh	⊢ ( 𝐴 ∈ S_ℋ → ( ⊥ ‘ 𝐴 ) ∈ S_ℋ )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐴 ∈ S_ℋ → 𝐴 ⊆ ℋ )
2		ocsh	⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ∈ S_ℋ )
3	1 2	syl	⊢ ( 𝐴 ∈ S_ℋ → ( ⊥ ‘ 𝐴 ) ∈ S_ℋ )