Metamath Proof Explorer


Theorem ge0divd

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 ) ) )

Proof

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 ) ) )