Theorem divge0d

Description: The ratio of nonnegative and positive numbers is nonnegative. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	|- ( ph -> A e. RR )
	rpgecld.2	|- ( ph -> B e. RR+ )
	divge0d.3	|- ( ph -> 0 <_ A )
Assertion	divge0d	|- ( ph -> 0 <_ ( A / B ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	|- ( ph -> A e. RR )
2		rpgecld.2	|- ( ph -> B e. RR+ )
3		divge0d.3	|- ( ph -> 0 <_ A )
4	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
5		divge0	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
6	1 3 4 5	syl21anc	|- ( ph -> 0 <_ ( A / B ) )