Metamath Proof Explorer


Theorem leadd1i

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 11-Aug-1999)

Ref Expression
Hypotheses lt2.1
|- A e. RR
lt2.2
|- B e. RR
lt2.3
|- C e. RR
Assertion leadd1i
|- ( A <_ B <-> ( A + C ) <_ ( B + C ) )

Proof

Step Hyp Ref Expression
1 lt2.1
 |-  A e. RR
2 lt2.2
 |-  B e. RR
3 lt2.3
 |-  C e. RR
4 leadd1
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) )
5 1 2 3 4 mp3an
 |-  ( A <_ B <-> ( A + C ) <_ ( B + C ) )