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Metamath Proof Explorer

Theorem suble0

Description: Nonpositive subtraction. (Contributed by NM, 20-Mar-2008) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion suble0 
|- ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> A <_ B ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 suble 
 |-  ( ( A e. RR /\ B e. RR /\ 0 e. RR ) -> ( ( A - B ) <_ 0 <-> ( A - 0 ) <_ B ) )
3 1 2 mp3an3 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> ( A - 0 ) <_ B ) )
4 simpl 
 |-  ( ( A e. RR /\ B e. RR ) -> A e. RR )
5 4 recnd 
 |-  ( ( A e. RR /\ B e. RR ) -> A e. CC )
6 5 subid1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A - 0 ) = A )
7 6 breq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A - 0 ) <_ B <-> A <_ B ) )
8 3 7 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> A <_ B ) )