Metamath Proof Explorer


Theorem ltrec1d

Description: Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpred.1
|- ( ph -> A e. RR+ )
rpaddcld.1
|- ( ph -> B e. RR+ )
ltrec1d.2
|- ( ph -> ( 1 / A ) < B )
Assertion ltrec1d
|- ( ph -> ( 1 / B ) < A )

Proof

Step Hyp Ref Expression
1 rpred.1
 |-  ( ph -> A e. RR+ )
2 rpaddcld.1
 |-  ( ph -> B e. RR+ )
3 ltrec1d.2
 |-  ( ph -> ( 1 / A ) < B )
4 1 rpregt0d
 |-  ( ph -> ( A e. RR /\ 0 < A ) )
5 2 rpregt0d
 |-  ( ph -> ( B e. RR /\ 0 < B ) )
6 ltrec1
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) )
7 4 5 6 syl2anc
 |-  ( ph -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) )
8 3 7 mpbid
 |-  ( ph -> ( 1 / B ) < A )