Theorem shoccl

Description: Closure of complement of Hilbert subspace. Part of Remark 3.12 of Beran p. 107. (Contributed by NM, 13-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	shoccl	⊢ ( 𝐴 ∈ Sℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )

Proof

Step	Hyp	Ref	Expression
1		shss	⊢ ( 𝐴 ∈ Sℋ → 𝐴 ⊆ ℋ )
2		occl	⊢ ( 𝐴 ⊆ ℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
3	1 2	syl	⊢ ( 𝐴 ∈ Sℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )