Description: Addition to both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | leidd.1 | |- ( ph -> A e. RR ) |
|
ltnegd.2 | |- ( ph -> B e. RR ) |
||
ltadd1d.3 | |- ( ph -> C e. RR ) |
||
leadd1dd.4 | |- ( ph -> A <_ B ) |
||
Assertion | leadd2dd | |- ( ph -> ( C + A ) <_ ( C + B ) ) |
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 | leadd1dd.4 | |- ( ph -> A <_ B ) |
|
5 | 1 2 3 | leadd2d | |- ( ph -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) ) |
6 | 4 5 | mpbid | |- ( ph -> ( C + A ) <_ ( C + B ) ) |