Theorem xleneg

Description: Extended real version of leneg . (Contributed by Mario Carneiro, 20-Aug-2015)

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

Proof

Step	Hyp	Ref	Expression
1		xltneg	|- ( ( B e. RR* /\ A e. RR* ) -> ( B < A <-> -e A < -e B ) )
2	1	ancoms	|- ( ( A e. RR* /\ B e. RR* ) -> ( B < A <-> -e A < -e B ) )
3	2	notbid	|- ( ( A e. RR* /\ B e. RR* ) -> ( -. B < A <-> -. -e A < -e B ) )
4		xrlenlt	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -. B < A ) )
5		xnegcl	|- ( B e. RR* -> -e B e. RR* )
6		xnegcl	|- ( A e. RR* -> -e A e. RR* )
7		xrlenlt	|- ( ( -e B e. RR* /\ -e A e. RR* ) -> ( -e B <_ -e A <-> -. -e A < -e B ) )
8	5 6 7	syl2anr	|- ( ( A e. RR* /\ B e. RR* ) -> ( -e B <_ -e A <-> -. -e A < -e B ) )
9	3 4 8	3bitr4d	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> -e B <_ -e A ) )