Description: Lemma to show a nonnegative number is zero. (Contributed by NM, 8-Oct-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lt2.1 | |- A e. RR |
|
lt2.2 | |- B e. RR |
||
Assertion | lesub0i | |- ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lt2.1 | |- A e. RR |
|
2 | lt2.2 | |- B e. RR |
|
3 | lesub0 | |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) ) |
|
4 | 1 2 3 | mp2an | |- ( ( 0 <_ A /\ B <_ ( B - A ) ) <-> A = 0 ) |