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Metamath Proof Explorer

Theorem lenegcon1

Description: Contraposition of negative in 'less than or equal to'. (Contributed by NM, 10-May-2004)

Ref Expression
Assertion lenegcon1 
|- ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ A ) )

Proof

Step Hyp Ref Expression
1 renegcl 
 |-  ( A e. RR -> -u A e. RR )
2 leneg 
 |-  ( ( -u A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ -u -u A ) )
3 1 2 sylan 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ -u -u A ) )
4 recn 
 |-  ( A e. RR -> A e. CC )
5 4 negnegd 
 |-  ( A e. RR -> -u -u A = A )
6 5 breq2d 
 |-  ( A e. RR -> ( -u B <_ -u -u A <-> -u B <_ A ) )
7 6 adantr 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u B <_ -u -u A <-> -u B <_ A ) )
8 3 7 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A <_ B <-> -u B <_ A ) )