Theorem probvalrnd

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

	Ref	Expression
Hypotheses	probmeasd.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
	probvalrnd.1	⊢ ( 𝜑 → 𝐴 ∈ dom 𝑃 )
Assertion	probvalrnd	⊢ ( 𝜑 → ( 𝑃 ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		probmeasd.1	⊢ ( 𝜑 → 𝑃 ∈ Prob )
2		probvalrnd.1	⊢ ( 𝜑 → 𝐴 ∈ dom 𝑃 )
3		unitssre	⊢ ( 0 [,] 1 ) ⊆ ℝ
4		prob01	⊢ ( ( 𝑃 ∈ Prob ∧ 𝐴 ∈ dom 𝑃 ) → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) )
5	1 2 4	syl2anc	⊢ ( 𝜑 → ( 𝑃 ‘ 𝐴 ) ∈ ( 0 [,] 1 ) )
6	3 5	sselid	⊢ ( 𝜑 → ( 𝑃 ‘ 𝐴 ) ∈ ℝ )