Theorem volioo

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

	Ref	Expression
Assertion	volioo	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )

Proof

Step	Hyp	Ref	Expression
1		ioombl	|- ( A (,) B ) e. dom vol
2		mblvol	|- ( ( A (,) B ) e. dom vol -> ( vol ` ( A (,) B ) ) = ( vol* ` ( A (,) B ) ) )
3	1 2	ax-mp	|- ( vol ` ( A (,) B ) ) = ( vol* ` ( A (,) B ) )
4		ovolioo	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol* ` ( A (,) B ) ) = ( B - A ) )
5	3 4	eqtrid	|- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( vol ` ( A (,) B ) ) = ( B - A ) )