Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 2-Aug-1999)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ltplus1.1 | |- A e. RR |
|
prodgt0.2 | |- B e. RR |
||
ltmul1.3 | |- C e. RR |
||
Assertion | lemul1i | |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltplus1.1 | |- A e. RR |
|
2 | prodgt0.2 | |- B e. RR |
|
3 | ltmul1.3 | |- C e. RR |
|
4 | lemul1 | |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) |
|
5 | 1 2 4 | mp3an12 | |- ( ( C e. RR /\ 0 < C ) -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) |
6 | 3 5 | mpan | |- ( 0 < C -> ( A <_ B <-> ( A x. C ) <_ ( B x. C ) ) ) |