Metamath Proof Explorer


Theorem lesubaddd

Description: 'Less than or equal to' relationship between subtraction and addition. (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 lesubaddd
|- ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )

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 lesubadd
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( ( A - B ) <_ C <-> A <_ ( C + B ) ) )