Theorem lemul12ad

Description: Comparison of product of two nonnegative numbers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	|- ( ph -> A e. RR )
	divgt0d.2	|- ( ph -> B e. RR )
	lemul1ad.3	|- ( ph -> C e. RR )
	ltmul12ad.3	|- ( ph -> D e. RR )
	lemul12ad.4	|- ( ph -> 0 <_ A )
	lemul12ad.5	|- ( ph -> 0 <_ C )
	lemul12ad.6	|- ( ph -> A <_ B )
	lemul12ad.7	|- ( ph -> C <_ D )
Assertion	lemul12ad	|- ( ph -> ( A x. C ) <_ ( B x. D ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	|- ( ph -> A e. RR )
2		divgt0d.2	|- ( ph -> B e. RR )
3		lemul1ad.3	|- ( ph -> C e. RR )
4		ltmul12ad.3	|- ( ph -> D e. RR )
5		lemul12ad.4	|- ( ph -> 0 <_ A )
6		lemul12ad.5	|- ( ph -> 0 <_ C )
7		lemul12ad.6	|- ( ph -> A <_ B )
8		lemul12ad.7	|- ( ph -> C <_ D )
9	1 5	jca	|- ( ph -> ( A e. RR /\ 0 <_ A ) )
10	3 6	jca	|- ( ph -> ( C e. RR /\ 0 <_ C ) )
11		lemul12a	|- ( ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR ) /\ ( ( C e. RR /\ 0 <_ C ) /\ D e. RR ) ) -> ( ( A <_ B /\ C <_ D ) -> ( A x. C ) <_ ( B x. D ) ) )
12	9 2 10 4 11	syl22anc	|- ( ph -> ( ( A <_ B /\ C <_ D ) -> ( A x. C ) <_ ( B x. D ) ) )
13	7 8 12	mp2and	|- ( ph -> ( A x. C ) <_ ( B x. D ) )