Description: Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | leidd.1 | |- ( ph -> A e. RR ) |
|
ltnegd.2 | |- ( ph -> B e. RR ) |
||
Assertion | suble0d | |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | |- ( ph -> A e. RR ) |
|
2 | ltnegd.2 | |- ( ph -> B e. RR ) |
|
3 | suble0 | |- ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> A <_ B ) ) |
|
4 | 1 2 3 | syl2anc | |- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) ) |