Metamath Proof Explorer


Theorem ltadd1dd

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
ltadd1d.3
|- ( ph -> C e. RR )
ltadd1dd.4
|- ( ph -> A < B )
Assertion ltadd1dd
|- ( ph -> ( A + C ) < ( B + C ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 ltadd1d.3
 |-  ( ph -> C e. RR )
4 ltadd1dd.4
 |-  ( ph -> A < B )
5 1 2 3 ltadd1d
 |-  ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )
6 4 5 mpbid
 |-  ( ph -> ( A + C ) < ( B + C ) )