Theorem volicc

Description: The Lebesgue measure of a closed interval. (Contributed by Glauco Siliprandi, 8-Apr-2021)

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

Proof

Step	Hyp	Ref	Expression
1		iccmbl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ∈ dom vol )
2		mblvol	⊢ ( ( 𝐴 [,] 𝐵 ) ∈ dom vol → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) )
4	3	3adant3	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) )
5		ovolicc	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol* ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) )
6	4 5	eqtrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ) → ( vol ‘ ( 𝐴 [,] 𝐵 ) ) = ( 𝐵 − 𝐴 ) )