Metamath Proof Explorer

Theorem lemin

Description: Two ways of saying a number is less than or equal to the minimum of two others. (Contributed by NM, 3-Aug-2007)

Ref Expression
Assertion lemin 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ if ( B <_ C , B , C ) <-> ( A <_ B /\ A <_ C ) ) )

Proof

Step Hyp Ref Expression
1 rexr 
 |-  ( A e. RR -> A e. RR* )
2 rexr 
 |-  ( B e. RR -> B e. RR* )
3 rexr 
 |-  ( C e. RR -> C e. RR* )
4 xrlemin 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ if ( B <_ C , B , C ) <-> ( A <_ B /\ A <_ C ) ) )
5 1 2 3 4 syl3an 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ if ( B <_ C , B , C ) <-> ( A <_ B /\ A <_ C ) ) )