Theorem ltmulgt12d

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

	Ref	Expression
Hypotheses	rpgecld.1	|- ( ph -> A e. RR )
	rpgecld.2	|- ( ph -> B e. RR+ )
Assertion	ltmulgt12d	|- ( ph -> ( 1 < A <-> B < ( A x. B ) ) )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	|- ( ph -> A e. RR )
2		rpgecld.2	|- ( ph -> B e. RR+ )
3	2	rpred	|- ( ph -> B e. RR )
4	2	rpgt0d	|- ( ph -> 0 < B )
5		ltmulgt12	|- ( ( B e. RR /\ A e. RR /\ 0 < B ) -> ( 1 < A <-> B < ( A x. B ) ) )
6	3 1 4 5	syl3anc	|- ( ph -> ( 1 < A <-> B < ( A x. B ) ) )