Theorem ledivdivd

Description: Invert ratios of positive numbers and swap their ordering. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	|- ( ph -> A e. RR+ )
	rpaddcld.1	|- ( ph -> B e. RR+ )
	ltdiv2d.3	|- ( ph -> C e. RR+ )
	ledivdivd.4	|- ( ph -> D e. RR+ )
	ledivdivd.5	|- ( ph -> ( A / B ) <_ ( C / D ) )
Assertion	ledivdivd	|- ( ph -> ( D / C ) <_ ( B / A ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2		rpaddcld.1	|- ( ph -> B e. RR+ )
3		ltdiv2d.3	|- ( ph -> C e. RR+ )
4		ledivdivd.4	|- ( ph -> D e. RR+ )
5		ledivdivd.5	|- ( ph -> ( A / B ) <_ ( C / D ) )
6	1	rpregt0d	|- ( ph -> ( A e. RR /\ 0 < A ) )
7	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
8	3	rpregt0d	|- ( ph -> ( C e. RR /\ 0 < C ) )
9	4	rpregt0d	|- ( ph -> ( D e. RR /\ 0 < D ) )
10		ledivdiv	|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) )
11	6 7 8 9 10	syl22anc	|- ( ph -> ( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) )
12	5 11	mpbid	|- ( ph -> ( D / C ) <_ ( B / A ) )