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Metamath Proof Explorer

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 ) )