Metamath Proof Explorer


Theorem ltdiv2d

Description: Division of a positive number by both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpred.1
|- ( ph -> A e. RR+ )
rpaddcld.1
|- ( ph -> B e. RR+ )
ltdiv2d.3
|- ( ph -> C e. RR+ )
Assertion ltdiv2d
|- ( ph -> ( A < B <-> ( C / B ) < ( C / A ) ) )

Proof

Step Hyp Ref Expression
1 rpred.1
 |-  ( ph -> A e. RR+ )
2 rpaddcld.1
 |-  ( ph -> B e. RR+ )
3 ltdiv2d.3
 |-  ( ph -> C e. RR+ )
4 1 rpregt0d
 |-  ( ph -> ( A e. RR /\ 0 < A ) )
5 2 rpregt0d
 |-  ( ph -> ( B e. RR /\ 0 < B ) )
6 3 rpregt0d
 |-  ( ph -> ( C e. RR /\ 0 < C ) )
7 ltdiv2
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C / B ) < ( C / A ) ) )
8 4 5 6 7 syl3anc
 |-  ( ph -> ( A < B <-> ( C / B ) < ( C / A ) ) )