Metamath Proof Explorer

Theorem subge02d

Description: Nonnegative subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 
|- ( ph -> A e. RR )
ltnegd.2 
|- ( ph -> B e. RR )
Assertion subge02d 
|- ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) )

Proof

Step Hyp Ref Expression
1 leidd.1 
 |-  ( ph -> A e. RR )
2 ltnegd.2 
 |-  ( ph -> B e. RR )
3 subge02 
 |-  ( ( A e. RR /\ B e. RR ) -> ( 0 <_ B <-> ( A - B ) <_ A ) )
4 1 2 3 syl2anc 
 |-  ( ph -> ( 0 <_ B <-> ( A - B ) <_ A ) )