Theorem ltnegcon2

Description: Contraposition of negative in 'less than'. (Contributed by Mario Carneiro, 25-Feb-2015)

	Ref	Expression
Assertion	ltnegcon2	|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )

Proof

Step	Hyp	Ref	Expression
1		renegcl	|- ( B e. RR -> -u B e. RR )
2		ltneg	|- ( ( A e. RR /\ -u B e. RR ) -> ( A < -u B <-> -u -u B < -u A ) )
3	1 2	sylan2	|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> -u -u B < -u A ) )
4		simpr	|- ( ( A e. RR /\ B e. RR ) -> B e. RR )
5	4	recnd	|- ( ( A e. RR /\ B e. RR ) -> B e. CC )
6	5	negnegd	|- ( ( A e. RR /\ B e. RR ) -> -u -u B = B )
7	6	breq1d	|- ( ( A e. RR /\ B e. RR ) -> ( -u -u B < -u A <-> B < -u A ) )
8	3 7	bitrd	|- ( ( A e. RR /\ B e. RR ) -> ( A < -u B <-> B < -u A ) )