Description: Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005)
Ref | Expression | ||
---|---|---|---|
Assertion | ltaddpos2 | |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltaddpos | |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( B + A ) ) ) |
|
2 | recn | |- ( A e. RR -> A e. CC ) |
|
3 | recn | |- ( B e. RR -> B e. CC ) |
|
4 | addcom | |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) ) |
|
5 | 2 3 4 | syl2an | |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) ) |
6 | 5 | breq2d | |- ( ( A e. RR /\ B e. RR ) -> ( B < ( A + B ) <-> B < ( B + A ) ) ) |
7 | 1 6 | bitr4d | |- ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) ) |