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Metamath Proof Explorer

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 ) )