Description: Division of a nonnegative number by a positive number. (Contributed by Mario Carneiro, 28-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rpgecld.1 | |- ( ph -> A e. RR ) |
|
rpgecld.2 | |- ( ph -> B e. RR+ ) |
||
Assertion | ge0divd | |- ( ph -> ( 0 <_ A <-> 0 <_ ( A / B ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgecld.1 | |- ( ph -> A e. RR ) |
|
2 | rpgecld.2 | |- ( ph -> B e. RR+ ) |
|
3 | 2 | rpred | |- ( ph -> B e. RR ) |
4 | 2 | rpgt0d | |- ( ph -> 0 < B ) |
5 | ge0div | |- ( ( A e. RR /\ B e. RR /\ 0 < B ) -> ( 0 <_ A <-> 0 <_ ( A / B ) ) ) |
|
6 | 1 3 4 5 | syl3anc | |- ( ph -> ( 0 <_ A <-> 0 <_ ( A / B ) ) ) |