Theorem lesub

Description: Swap subtrahends in an inequality. (Contributed by NM, 29-Sep-2005) (Proof shortened by Andrew Salmon, 19-Nov-2011)

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

Proof

Step	Hyp	Ref	Expression
1		leaddsub	|- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A + C ) <_ B <-> A <_ ( B - C ) ) )
2		leaddsub2	|- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( ( A + C ) <_ B <-> C <_ ( B - A ) ) )
3	1 2	bitr3d	|- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A <_ ( B - C ) <-> C <_ ( B - A ) ) )
4	3	3com23	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ ( B - C ) <-> C <_ ( B - A ) ) )