Theorem lediv2ad

Description: Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpred.1	|- ( ph -> A e. RR+ )
	rpaddcld.1	|- ( ph -> B e. RR+ )
	lediv2ad.3	|- ( ph -> C e. RR )
	lediv2ad.4	|- ( ph -> 0 <_ C )
	lediv2ad.5	|- ( ph -> A <_ B )
Assertion	lediv2ad	|- ( ph -> ( C / B ) <_ ( C / A ) )

Proof

Step	Hyp	Ref	Expression
1		rpred.1	|- ( ph -> A e. RR+ )
2		rpaddcld.1	|- ( ph -> B e. RR+ )
3		lediv2ad.3	|- ( ph -> C e. RR )
4		lediv2ad.4	|- ( ph -> 0 <_ C )
5		lediv2ad.5	|- ( ph -> A <_ B )
6	1	rpregt0d	|- ( ph -> ( A e. RR /\ 0 < A ) )
7	2	rpregt0d	|- ( ph -> ( B e. RR /\ 0 < B ) )
8	3 4	jca	|- ( ph -> ( C e. RR /\ 0 <_ C ) )
9		lediv2a	|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( C / B ) <_ ( C / A ) )
10	6 7 8 5 9	syl31anc	|- ( ph -> ( C / B ) <_ ( C / A ) )