Theorem xlenegcon2

Description: Extended real version of lenegcon2 . (Contributed by Glauco Siliprandi, 23-Apr-2023)

	Ref	Expression
Assertion	xlenegcon2	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ -e B <-> B <_ -e A ) )

Proof

Step	Hyp	Ref	Expression
1		xnegcl	|- ( B e. RR* -> -e B e. RR* )
2		xleneg	|- ( ( A e. RR* /\ -e B e. RR* ) -> ( A <_ -e B <-> -e -e B <_ -e A ) )
3	1 2	sylan2	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ -e B <-> -e -e B <_ -e A ) )
4		xnegneg	|- ( B e. RR* -> -e -e B = B )
5	4	breq1d	|- ( B e. RR* -> ( -e -e B <_ -e A <-> B <_ -e A ) )
6	5	adantl	|- ( ( A e. RR* /\ B e. RR* ) -> ( -e -e B <_ -e A <-> B <_ -e A ) )
7	3 6	bitrd	|- ( ( A e. RR* /\ B e. RR* ) -> ( A <_ -e B <-> B <_ -e A ) )