Metamath Proof Explorer

Theorem addgt0i

Description: Addition of 2 positive numbers is positive. (Contributed by NM, 16-May-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)

Ref Expression
Hypotheses lt2.1 
|- A e. RR
lt2.2 
|- B e. RR
Assertion addgt0i 
|- ( ( 0 < A /\ 0 < B ) -> 0 < ( A + B ) )

Proof

Step Hyp Ref Expression
1 lt2.1 
 |-  A e. RR
2 lt2.2 
 |-  B e. RR
3 addgt0 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < B ) ) -> 0 < ( A + B ) )
4 1 2 3 mpanl12 
 |-  ( ( 0 < A /\ 0 < B ) -> 0 < ( A + B ) )