Theorem choccl

Description: Closure of complement of Hilbert subspace. Part of Remark 3.12 of Beran p. 107. (Contributed by NM, 22-Jul-2001) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		chsh	⊢ ( 𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		shoccl	⊢ ( 𝐴 ∈ Sℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )
3	1 2	syl	⊢ ( 𝐴 ∈ Cℋ → ( ⊥ ‘ 𝐴 ) ∈ Cℋ )