Description: Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | xleadd2d.1 | |- ( ph -> A e. RR* ) |
|
xleadd2d.2 | |- ( ph -> B e. RR* ) |
||
xleadd2d.3 | |- ( ph -> C e. RR* ) |
||
xleadd2d.4 | |- ( ph -> A <_ B ) |
||
Assertion | xleadd2d | |- ( ph -> ( C +e A ) <_ ( C +e B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xleadd2d.1 | |- ( ph -> A e. RR* ) |
|
2 | xleadd2d.2 | |- ( ph -> B e. RR* ) |
|
3 | xleadd2d.3 | |- ( ph -> C e. RR* ) |
|
4 | xleadd2d.4 | |- ( ph -> A <_ B ) |
|
5 | xleadd2a | |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( C +e A ) <_ ( C +e B ) ) |
|
6 | 1 2 3 4 5 | syl31anc | |- ( ph -> ( C +e A ) <_ ( C +e B ) ) |