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 ) ) |
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 ) ) |