Theorem lenegcon2

Description: Contraposition of negative in 'less than or equal to'. (Contributed by NM, 8-Oct-2005)

	Ref	Expression
Assertion	lenegcon2	|- ( ( 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		leneg	|- ( ( 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		recn	|- ( B e. RR -> B e. CC )
5	4	negnegd	|- ( B e. RR -> -u -u B = B )
6	5	adantl	|- ( ( 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 ) )