Theorem probvalrnd

Description: The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017)

	Ref	Expression
Hypotheses	probmeasd.1	|- ( ph -> P e. Prob )
	probvalrnd.1	|- ( ph -> A e. dom P )
Assertion	probvalrnd	|- ( ph -> ( P ` A ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		probmeasd.1	|- ( ph -> P e. Prob )
2		probvalrnd.1	|- ( ph -> A e. dom P )
3		unitssre	|- ( 0 [,] 1 ) C_ RR
4		prob01	|- ( ( P e. Prob /\ A e. dom P ) -> ( P ` A ) e. ( 0 [,] 1 ) )
5	1 2 4	syl2anc	|- ( ph -> ( P ` A ) e. ( 0 [,] 1 ) )
6	3 5	sseldi	|- ( ph -> ( P ` A ) e. RR )