spanval

Description: Value of the linear span of a subset of Hilbert space. The span is the intersection of all subspaces constraining the subset. Definition of span in Schechter p. 276. (Contributed by NM, 2-Jun-2004) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	spanval	$⊢ A \subseteq ℋ \to span (A) = ⋂ \{x \in S_{ℋ} \| A \subseteq x\}$

Proof

Step	Hyp	Ref	Expression
1		df-span	$⊢ span = (y \in 𝒫 ℋ ⟼ ⋂ \{x \in S_{ℋ} \| y \subseteq x\})$
2		sseq1	$⊢ y = A \to (y \subseteq x \leftrightarrow A \subseteq x)$
3	2	rabbidv	$⊢ y = A \to \{x \in S_{ℋ} \| y \subseteq x\} = \{x \in S_{ℋ} \| A \subseteq x\}$
4	3	inteqd	$⊢ y = A \to ⋂ \{x \in S_{ℋ} \| y \subseteq x\} = ⋂ \{x \in S_{ℋ} \| A \subseteq x\}$
5		ax-hilex	$⊢ ℋ \in V$
6	5	elpw2	$⊢ A \in 𝒫 ℋ \leftrightarrow A \subseteq ℋ$
7	6	biimpri	$⊢ A \subseteq ℋ \to A \in 𝒫 ℋ$
8		helsh	$⊢ ℋ \in S_{ℋ}$
9		sseq2	$⊢ x = ℋ \to (A \subseteq x \leftrightarrow A \subseteq ℋ)$
10	9	rspcev	$⊢ (ℋ \in S_{ℋ} \land A \subseteq ℋ) \to \exists x \in S_{ℋ} A \subseteq x$
11	8 10	mpan	$⊢ A \subseteq ℋ \to \exists x \in S_{ℋ} A \subseteq x$
12		intexrab	$⊢ \exists x \in S_{ℋ} A \subseteq x \leftrightarrow ⋂ \{x \in S_{ℋ} \| A \subseteq x\} \in V$
13	11 12	sylib	$⊢ A \subseteq ℋ \to ⋂ \{x \in S_{ℋ} \| A \subseteq x\} \in V$
14	1 4 7 13	fvmptd3	$⊢ A \subseteq ℋ \to span (A) = ⋂ \{x \in S_{ℋ} \| A \subseteq x\}$

Metamath Proof Explorer

Theorem spanval

Proof