Metamath Proof Explorer


Theorem ltaddpos2d

Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
Assertion ltaddpos2d
|- ( ph -> ( 0 < A <-> B < ( A + B ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 ltaddpos2
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 < A <-> B < ( A + B ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( 0 < A <-> B < ( A + B ) ) )