Theorem voncl

Description: The Lebesgue measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypotheses	voncl.x	|- ( ph -> X e. Fin )
	voncl.s	|- S = dom ( voln ` X )
	voncl.a	|- ( ph -> A e. S )
Assertion	voncl	|- ( ph -> ( ( voln ` X ) ` A ) e. ( 0 [,] +oo ) )

Proof

Step	Hyp	Ref	Expression
1		voncl.x	|- ( ph -> X e. Fin )
2		voncl.s	|- S = dom ( voln ` X )
3		voncl.a	|- ( ph -> A e. S )
4	1	vonmea	|- ( ph -> ( voln ` X ) e. Meas )
5	4 2 3	meacl	|- ( ph -> ( ( voln ` X ) ` A ) e. ( 0 [,] +oo ) )