Metamath Proof Explorer


Theorem lenegd

Description: Negative of both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
Assertion lenegd
|- ( ph -> ( A <_ B <-> -u B <_ -u A ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 leneg
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -u B <_ -u A ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( A <_ B <-> -u B <_ -u A ) )