Description: Addition to both sides of 'less than'. (Contributed by Mario Carneiro, 27-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ltd.1 | |- ( ph -> A e. RR ) |
|
ltd.2 | |- ( ph -> B e. RR ) |
||
letrd.3 | |- ( ph -> C e. RR ) |
||
Assertion | ltadd2d | |- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | |- ( ph -> A e. RR ) |
|
2 | ltd.2 | |- ( ph -> B e. RR ) |
|
3 | letrd.3 | |- ( ph -> C e. RR ) |
|
4 | ltadd2 | |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) ) |
|
5 | 1 2 3 4 | syl3anc | |- ( ph -> ( A < B <-> ( C + A ) < ( C + B ) ) ) |