Description: Addition of extended reals preserves the "less than or equal to" relation, in the left slot. (Contributed by Glauco Siliprandi, 17-Aug-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | xleadd1d.1 | |- ( ph -> A e. RR* ) |
|
| xleadd1d.2 | |- ( ph -> B e. RR* ) |
||
| xleadd1d.3 | |- ( ph -> C e. RR* ) |
||
| xleadd1d.4 | |- ( ph -> A <_ B ) |
||
| Assertion | xleadd1d | |- ( ph -> ( A +e C ) <_ ( B +e C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xleadd1d.1 | |- ( ph -> A e. RR* ) |
|
| 2 | xleadd1d.2 | |- ( ph -> B e. RR* ) |
|
| 3 | xleadd1d.3 | |- ( ph -> C e. RR* ) |
|
| 4 | xleadd1d.4 | |- ( ph -> A <_ B ) |
|
| 5 | xleadd1a | |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( A +e C ) <_ ( B +e C ) ) |
|
| 6 | 1 2 3 4 5 | syl31anc | |- ( ph -> ( A +e C ) <_ ( B +e C ) ) |