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 )