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 ) ) )