Description: The sum of nonnegative and positive numbers is positive. (Contributed by NM, 28-Dec-2005) (Proof shortened by Andrew Salmon, 19-Nov-2011)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | addgtge0 | |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> 0 < ( A + B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 00id | |- ( 0 + 0 ) = 0 |
|
| 2 | 0re | |- 0 e. RR |
|
| 3 | ltleadd | |- ( ( ( 0 e. RR /\ 0 e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( ( 0 < A /\ 0 <_ B ) -> ( 0 + 0 ) < ( A + B ) ) ) |
|
| 4 | 2 2 3 | mpanl12 | |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 <_ B ) -> ( 0 + 0 ) < ( A + B ) ) ) |
| 5 | 4 | imp | |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> ( 0 + 0 ) < ( A + B ) ) |
| 6 | 1 5 | eqbrtrrid | |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 <_ B ) ) -> 0 < ( A + B ) ) |