Metamath Proof Explorer


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 ) )