Description: When a subtraction gives a negative result. (Contributed by Glauco Siliprandi, 11-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | sublt0d.1 | |- ( ph -> A e. RR ) |
|
sublt0d.2 | |- ( ph -> B e. RR ) |
||
Assertion | sublt0d | |- ( ph -> ( ( A - B ) < 0 <-> A < B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sublt0d.1 | |- ( ph -> A e. RR ) |
|
2 | sublt0d.2 | |- ( ph -> B e. RR ) |
|
3 | 0red | |- ( ph -> 0 e. RR ) |
|
4 | 1 2 3 | ltsubaddd | |- ( ph -> ( ( A - B ) < 0 <-> A < ( 0 + B ) ) ) |
5 | 2 | recnd | |- ( ph -> B e. CC ) |
6 | 5 | addid2d | |- ( ph -> ( 0 + B ) = B ) |
7 | 6 | breq2d | |- ( ph -> ( A < ( 0 + B ) <-> A < B ) ) |
8 | 4 7 | bitrd | |- ( ph -> ( ( A - B ) < 0 <-> A < B ) ) |