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Metamath Proof Explorer

Theorem sshjval2

Description: Value of join in the set of closed subspaces of Hilbert space CH . (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	sshjval2	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⋂ \{x \in C_{ℋ} \| A \cup B \subseteq x\}$

Proof

Step	Hyp	Ref	Expression
1		sshjval	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$
2		unss	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \leftrightarrow A \cup B \subseteq ℋ$
3		ococin	$⊢ A \cup B \subseteq ℋ \to ⊥ (⊥ (A \cup B)) = ⋂ \{x \in C_{ℋ} \| A \cup B \subseteq x\}$
4	2 3	sylbi	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to ⊥ (⊥ (A \cup B)) = ⋂ \{x \in C_{ℋ} \| A \cup B \subseteq x\}$
5	1 4	eqtrd	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⋂ \{x \in C_{ℋ} \| A \cup B \subseteq x\}$