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Metamath Proof Explorer

Theorem iblconstmpt

Description: A constant function is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019)

		Ref	Expression
	Assertion	iblconstmpt	$⊢ (A \in dom vol \land vol (A) \in ℝ \land B \in ℂ) \to (x \in A ⟼ B) \in 𝐿^{1}$

Step	Hyp	Ref	Expression
1		fconstmpt	$⊢ A \times \{B\} = (x \in A ⟼ B)$
2		iblconst	$⊢ (A \in dom vol \land vol (A) \in ℝ \land B \in ℂ) \to A \times \{B\} \in 𝐿^{1}$
3	1 2	eqeltrrid	$⊢ (A \in dom vol \land vol (A) \in ℝ \land B \in ℂ) \to (x \in A ⟼ B) \in 𝐿^{1}$