Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (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 ) |
||
ltadd1dd.4 | |- ( ph -> A < B ) |
||
Assertion | ltadd1dd | |- ( ph -> ( 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 | ltadd1dd.4 | |- ( ph -> A < B ) |
|
5 | 1 2 3 | ltadd1d | |- ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) ) |
6 | 4 5 | mpbid | |- ( ph -> ( A + C ) < ( B + C ) ) |