Description: Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | leidd.1 | |- ( ph -> A e. RR ) |
|
ltnegd.2 | |- ( ph -> B e. RR ) |
||
ltadd1d.3 | |- ( ph -> C e. RR ) |
||
Assertion | leadd1d | |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leidd.1 | |- ( ph -> A e. RR ) |
|
2 | ltnegd.2 | |- ( ph -> B e. RR ) |
|
3 | ltadd1d.3 | |- ( ph -> C e. RR ) |
|
4 | leadd1 | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) |
|
5 | 1 2 3 4 | syl3anc | |- ( ph -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) ) |