ovolcl

Description: The volume of a set is an extended real number. (Contributed by Mario Carneiro, 16-Mar-2014)

		Ref	Expression
	Assertion	ovolcl	$⊢ A \subseteq ℝ \to {vol}^{} (A) \in ℝ^{}$

Step	Hyp	Ref	Expression
1		eqid	$⊢ \{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} = \{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\}$
2	1	ovolval	$⊢ A \subseteq ℝ \to {vol}^{} (A) = sup (\{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} ℝ^{} <)$
3		ssrab2	$⊢ \{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} \subseteq ℝ^{*}$
4		infxrcl	$⊢ \{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} \subseteq ℝ^{} \to sup (\{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} ℝ^{} <) \in ℝ^{*}$
5	3 4	ax-mp	$⊢ sup (\{y \in ℝ^{} \| \exists f \in ({(\leq \cap ℝ^{2})}^{ℕ}) (A \subseteq ⋃ ran ((.) \circ f) \land y = sup (ran seq 1 (+ ((abs \circ -) \circ f)) ℝ^{} <))\} ℝ^{} <) \in ℝ^{}$
6	2 5	eqeltrdi	$⊢ A \subseteq ℝ \to {vol}^{} (A) \in ℝ^{}$

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