Theorem ltmul12ad

Description: Comparison of product of two positive 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 )
	ltmul12ad.4	|- ( ph -> 0 <_ A )
	ltmul12ad.5	|- ( ph -> A < B )
	ltmul12ad.6	|- ( ph -> 0 <_ C )
	ltmul12ad.7	|- ( ph -> C < D )
Assertion	ltmul12ad	|- ( 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		ltmul12ad.4	|- ( ph -> 0 <_ A )
6		ltmul12ad.5	|- ( ph -> A < B )
7		ltmul12ad.6	|- ( ph -> 0 <_ C )
8		ltmul12ad.7	|- ( ph -> C < D )
9	1 2	jca	|- ( ph -> ( A e. RR /\ B e. RR ) )
10	5 6	jca	|- ( ph -> ( 0 <_ A /\ A < B ) )
11	3 4	jca	|- ( ph -> ( C e. RR /\ D e. RR ) )
12	7 8	jca	|- ( ph -> ( 0 <_ C /\ C < D ) )
13		ltmul12a	|- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( ( C e. RR /\ D e. RR ) /\ ( 0 <_ C /\ C < D ) ) ) -> ( A x. C ) < ( B x. D ) )
14	9 10 11 12 13	syl22anc	|- ( ph -> ( A x. C ) < ( B x. D ) )