Theorem lemulge12d

Description: Multiplication by a number greater than or equal to 1. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltp1d.1	|- ( ph -> A e. RR )
	divgt0d.2	|- ( ph -> B e. RR )
	lemulge11d.3	|- ( ph -> 0 <_ A )
	lemulge11d.4	|- ( ph -> 1 <_ B )
Assertion	lemulge12d	|- ( ph -> A <_ ( B x. A ) )

Proof

Step	Hyp	Ref	Expression
1		ltp1d.1	|- ( ph -> A e. RR )
2		divgt0d.2	|- ( ph -> B e. RR )
3		lemulge11d.3	|- ( ph -> 0 <_ A )
4		lemulge11d.4	|- ( ph -> 1 <_ B )
5		lemulge12	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 1 <_ B ) ) -> A <_ ( B x. A ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> A <_ ( B x. A ) )