Theorem rpgecld

Description: A number greater than or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	|- ( ph -> A e. RR )
	rpgecld.2	|- ( ph -> B e. RR+ )
	rpgecld.3	|- ( ph -> B <_ A )
Assertion	rpgecld	|- ( ph -> A e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	|- ( ph -> A e. RR )
2		rpgecld.2	|- ( ph -> B e. RR+ )
3		rpgecld.3	|- ( ph -> B <_ A )
4		rpgecl	|- ( ( B e. RR+ /\ A e. RR /\ B <_ A ) -> A e. RR+ )
5	2 1 3 4	syl3anc	|- ( ph -> A e. RR+ )