This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cvp

Description: The Hilbert lattice satisfies the covering property of Definition 7.4 of MaedaMaeda p. 31 and its converse. (Contributed by NM, 21-Jun-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	cvp	$⊢ (A \in C_{ℋ} \land B \in HAtoms) \to (A \cap B = 0_{ℋ} \leftrightarrow A ⋖_{ℋ} (A \lor_{ℋ} B))$

Proof

Step	Hyp	Ref	Expression
1		atelch	$⊢ B \in HAtoms \to B \in C_{ℋ}$
2		chincl	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to A \cap B \in C_{ℋ}$
3	1 2	sylan2	$⊢ (A \in C_{ℋ} \land B \in HAtoms) \to A \cap B \in C_{ℋ}$
4		atcveq0	$⊢ (A \cap B \in C_{ℋ} \land B \in HAtoms) \to ((A \cap B) ⋖_{ℋ} B \leftrightarrow A \cap B = 0_{ℋ})$
5	3 4	sylancom	$⊢ (A \in C_{ℋ} \land B \in HAtoms) \to ((A \cap B) ⋖_{ℋ} B \leftrightarrow A \cap B = 0_{ℋ})$
6		cvexch	$⊢ (A \in C_{ℋ} \land B \in C_{ℋ}) \to ((A \cap B) ⋖_{ℋ} B \leftrightarrow A ⋖_{ℋ} (A \lor_{ℋ} B))$
7	1 6	sylan2	$⊢ (A \in C_{ℋ} \land B \in HAtoms) \to ((A \cap B) ⋖_{ℋ} B \leftrightarrow A ⋖_{ℋ} (A \lor_{ℋ} B))$
8	5 7	bitr3d	$⊢ (A \in C_{ℋ} \land B \in HAtoms) \to (A \cap B = 0_{ℋ} \leftrightarrow A ⋖_{ℋ} (A \lor_{ℋ} B))$