Theorem mulgt0d

Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ltd.2	|- ( ph -> B e. RR )
	mulgt0d.3	|- ( ph -> 0 < A )
	mulgt0d.4	|- ( ph -> 0 < B )
Assertion	mulgt0d	|- ( ph -> 0 < ( A x. B ) )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		mulgt0d.3	|- ( ph -> 0 < A )
4		mulgt0d.4	|- ( ph -> 0 < B )
5		mulgt0	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
6	1 3 2 4 5	syl22anc	|- ( ph -> 0 < ( A x. B ) )