Metamath Proof Explorer


Theorem addgtge0d

Description: Addition of positive and nonnegative numbers is positive. (Contributed by Asger C. Ipsen, 12-May-2021)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
addgtge0d.3
|- ( ph -> 0 < A )
addgtge0d.4
|- ( ph -> 0 <_ B )
Assertion addgtge0d
|- ( ph -> 0 < ( A + B ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 addgtge0d.3
 |-  ( ph -> 0 < A )
4 addgtge0d.4
 |-  ( ph -> 0 <_ B )
5 addgtge0
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> 0 < ( A + B ) )
6 1 2 3 4 5 syl22anc
 |-  ( ph -> 0 < ( A + B ) )