Theorem volioo

Description: The measure of an open interval. (Contributed by Glauco Siliprandi, 29-Jun-2017)

	Ref	Expression
Assertion	volioo	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐵 − 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		ioombl	⊢ ( 𝐴 (,) 𝐵 ) ∈ dom vol
2		mblvol	⊢ ( ( 𝐴 (,) 𝐵 ) ∈ dom vol → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( vol* ‘ ( 𝐴 (,) 𝐵 ) ) )
3	1 2	ax-mp	⊢ ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( vol* ‘ ( 𝐴 (,) 𝐵 ) )
4		ovolioo	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol* ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐵 − 𝐴 ) )
5	3 4	syl5eq	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 (,) 𝐵 ) ) = ( 𝐵 − 𝐴 ) )