Metamath Proof Explorer


Theorem addge01d

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
Assertion addge01d
|- ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 addge01
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> A <_ ( A + B ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( 0 <_ B <-> A <_ ( A + B ) ) )