Metamath Proof Explorer

Theorem sshjval

Description: Value of join for subsets of Hilbert space. (Contributed by NM, 1-Nov-2000) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	sshjval	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$

Proof

Step	Hyp	Ref	Expression
1		ax-hilex	$⊢ ℋ \in V$
2	1	elpw2	$⊢ A \in 𝒫 ℋ \leftrightarrow A \subseteq ℋ$
3	1	elpw2	$⊢ B \in 𝒫 ℋ \leftrightarrow B \subseteq ℋ$
4		uneq12	$⊢ (x = A \land y = B) \to x \cup y = A \cup B$
5	4	fveq2d	$⊢ (x = A \land y = B) \to ⊥ (x \cup y) = ⊥ (A \cup B)$
6	5	fveq2d	$⊢ (x = A \land y = B) \to ⊥ (⊥ (x \cup y)) = ⊥ (⊥ (A \cup B))$
7		df-chj	$⊢ \lor_{ℋ} = (x \in 𝒫 ℋ, y \in 𝒫 ℋ ⟼ ⊥ (⊥ (x \cup y)))$
8		fvex	$⊢ ⊥ (⊥ (A \cup B)) \in V$
9	6 7 8	ovmpoa	$⊢ (A \in 𝒫 ℋ \land B \in 𝒫 ℋ) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$
10	2 3 9	syl2anbr	$⊢ (A \subseteq ℋ \land B \subseteq ℋ) \to A \lor_{ℋ} B = ⊥ (⊥ (A \cup B))$