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Metamath Proof Explorer

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 ∈ C_ℋ → ( ⊥ ‘ 𝐴 ) ∈ C_ℋ )

Proof

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