Metamath Proof Explorer


Theorem lerecd

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

Step Hyp Ref Expression
1 rpred.1
 |-  ( ph -> A e. RR+ )
2 rpaddcld.1
 |-  ( ph -> B e. RR+ )
3 1 rpregt0d
 |-  ( ph -> ( A e. RR /\ 0 < A ) )
4 2 rpregt0d
 |-  ( ph -> ( B e. RR /\ 0 < B ) )
5 lerec
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
6 3 4 5 syl2anc
 |-  ( ph -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )