Metamath Proof Explorer


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 ) ) )