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 ) )