Metamath Proof Explorer


Theorem ltadd2i

Description: Addition to both sides of 'less than'. (Contributed by NM, 21-Jan-1997) (Proof shortened by OpenAI, 25-Mar-2020)

Ref Expression
Hypotheses lt.1
|- A e. RR
lt.2
|- B e. RR
lt.3
|- C e. RR
Assertion ltadd2i
|- ( A < B <-> ( C + A ) < ( C + B ) )

Proof

Step Hyp Ref Expression
1 lt.1
 |-  A e. RR
2 lt.2
 |-  B e. RR
3 lt.3
 |-  C e. RR
4 ltadd2
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
5 1 2 3 4 mp3an
 |-  ( A < B <-> ( C + A ) < ( C + B ) )