Metamath Proof Explorer

Theorem maxle

Description: Two ways of saying the maximum of two numbers is less than or equal to a third. (Contributed by NM, 29-Sep-2005)

Ref Expression
Assertion maxle 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( if ( A <_ B , B , A ) <_ C <-> ( A <_ C /\ B <_ 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 xrmaxle 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( if ( A <_ B , B , A ) <_ C <-> ( A <_ C /\ B <_ C ) ) )
5 1 2 3 4 syl3an 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( if ( A <_ B , B , A ) <_ C <-> ( A <_ C /\ B <_ C ) ) )