Metamath Proof Explorer


Theorem lemul1ad

Description: Multiplication of both sides of 'less than or equal to' by a nonnegative number. (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 )
lemul1ad.4
|- ( ph -> 0 <_ C )
lemul1ad.5
|- ( ph -> A <_ B )
Assertion lemul1ad
|- ( ph -> ( A x. C ) <_ ( B x. C ) )

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 lemul1ad.4
 |-  ( ph -> 0 <_ C )
5 lemul1ad.5
 |-  ( ph -> A <_ B )
6 3 4 jca
 |-  ( ph -> ( C e. RR /\ 0 <_ C ) )
7 lemul1a
 |-  ( ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( A x. C ) <_ ( B x. C ) )
8 1 2 6 5 7 syl31anc
 |-  ( ph -> ( A x. C ) <_ ( B x. C ) )