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Metamath Proof Explorer

Theorem chocini

Description: Intersection of a closed subspace and its orthocomplement. Part of Proposition 1 of Kalmbach p. 65. (Contributed by NM, 11-Oct-1999) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	ch0le.1	$⊢ A \in C_{ℋ}$
	Assertion	chocini	$⊢ A \cap ⊥ (A) = 0_{ℋ}$

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	$⊢ A \in C_{ℋ}$
2	1	chshii	$⊢ A \in S_{ℋ}$
3		ocin	$⊢ A \in S_{ℋ} \to A \cap ⊥ (A) = 0_{ℋ}$
4	2 3	ax-mp	$⊢ A \cap ⊥ (A) = 0_{ℋ}$