Theorem lerec2d

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

	Ref	Expression
Hypotheses	rpred.1	|- ( ph -> A e. RR+ )
	rpaddcld.1	|- ( ph -> B e. RR+ )
	lerec2d.2	|- ( ph -> A <_ ( 1 / B ) )
Assertion	lerec2d	|- ( ph -> B <_ ( 1 / A ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2		rpaddcld.1	|- ( ph -> B e. RR+ )
3		lerec2d.2	|- ( ph -> A <_ ( 1 / B ) )
4	1	rpregt0d	|- ( ph -> ( A e. RR /\ 0 < A ) )
5	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
6		lerec2	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) )
7	4 5 6	syl2anc	|- ( ph -> ( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) )
8	3 7	mpbid	|- ( ph -> B <_ ( 1 / A ) )