Metamath Proof Explorer


Theorem ltdiv23d

Description: Swap denominator with other side of 'less than'. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses ltdiv23d.1
|- ( ph -> A e. RR )
ltdiv23d.2
|- ( ph -> B e. RR+ )
ltdiv23d.3
|- ( ph -> C e. RR+ )
ltdiv23d.4
|- ( ph -> ( A / B ) < C )
Assertion ltdiv23d
|- ( ph -> ( A / C ) < B )

Proof

Step Hyp Ref Expression
1 ltdiv23d.1
 |-  ( ph -> A e. RR )
2 ltdiv23d.2
 |-  ( ph -> B e. RR+ )
3 ltdiv23d.3
 |-  ( ph -> C e. RR+ )
4 ltdiv23d.4
 |-  ( ph -> ( A / B ) < C )
5 2 rpregt0d
 |-  ( ph -> ( B e. RR /\ 0 < B ) )
6 3 rpregt0d
 |-  ( ph -> ( C e. RR /\ 0 < C ) )
7 ltdiv23
 |-  ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) )
8 1 5 6 7 syl3anc
 |-  ( ph -> ( ( A / B ) < C <-> ( A / C ) < B ) )
9 4 8 mpbid
 |-  ( ph -> ( A / C ) < B )