Theorem xltneg

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

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

Proof

Step	Hyp	Ref	Expression
1		xltnegi	|- ( ( A e. RR* /\ B e. RR* /\ A < B ) -> -e B < -e A )
2	1	3expia	|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> -e B < -e A ) )
3		xnegcl	|- ( B e. RR* -> -e B e. RR* )
4		xnegcl	|- ( A e. RR* -> -e A e. RR* )
5		xltnegi	|- ( ( -e B e. RR* /\ -e A e. RR* /\ -e B < -e A ) -> -e -e A < -e -e B )
6	5	3expia	|- ( ( -e B e. RR* /\ -e A e. RR* ) -> ( -e B < -e A -> -e -e A < -e -e B ) )
7	3 4 6	syl2anr	|- ( ( A e. RR* /\ B e. RR* ) -> ( -e B < -e A -> -e -e A < -e -e B ) )
8		xnegneg	|- ( A e. RR* -> -e -e A = A )
9		xnegneg	|- ( B e. RR* -> -e -e B = B )
10	8 9	breqan12d	|- ( ( A e. RR* /\ B e. RR* ) -> ( -e -e A < -e -e B <-> A < B ) )
11	7 10	sylibd	|- ( ( A e. RR* /\ B e. RR* ) -> ( -e B < -e A -> A < B ) )
12	2 11	impbid	|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B <-> -e B < -e A ) )