Theorem gt0divd

Description: Division of a positive number by a positive number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	|- ( ph -> A e. RR )
	rpgecld.2	|- ( ph -> B e. RR+ )
Assertion	gt0divd	|- ( ph -> ( 0 < A <-> 0 < ( A / B ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	|- ( ph -> A e. RR )
2		rpgecld.2	|- ( ph -> B e. RR+ )
3	2	rpred	|- ( ph -> B e. RR )
4	2	rpgt0d	|- ( ph -> 0 < B )
5		gt0div	|- ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 < A <-> 0 < ( A / B ) ) )
6	1 3 4 5	syl3anc	|- ( ph -> ( 0 < A <-> 0 < ( A / B ) ) )