Metamath Proof Explorer


Theorem ltadd1d

Description: Addition to both sides of 'less than'. Theorem I.18 of Apostol p. 20. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
ltadd1d.3
|- ( ph -> C e. RR )
Assertion ltadd1d
|- ( ph -> ( A < B <-> ( 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 ltadd1
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A + C ) < ( B + C ) ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( A < B <-> ( A + C ) < ( B + C ) ) )