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Metamath Proof Explorer

Theorem lemaxle

Description: A real number which is less than or equal to a second real number is less than or equal to the maximum/supremum of the second real number and a third real number. (Contributed by AV, 8-Jun-2021)

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

Proof

Step Hyp Ref Expression
1 max2 
 |-  ( ( C e. RR /\ B e. RR ) -> B <_ if ( C <_ B , B , C ) )
2 1 ancoms 
 |-  ( ( B e. RR /\ C e. RR ) -> B <_ if ( C <_ B , B , C ) )
3 2 adantr 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> B <_ if ( C <_ B , B , C ) )
4 simpr 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> A e. RR )
5 simpll 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> B e. RR )
6 ifcl 
 |-  ( ( B e. RR /\ C e. RR ) -> if ( C <_ B , B , C ) e. RR )
7 6 adantr 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> if ( C <_ B , B , C ) e. RR )
8 letr 
 |-  ( ( A e. RR /\ B e. RR /\ if ( C <_ B , B , C ) e. RR ) -> ( ( A <_ B /\ B <_ if ( C <_ B , B , C ) ) -> A <_ if ( C <_ B , B , C ) ) )
9 4 5 7 8 syl3anc 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> ( ( A <_ B /\ B <_ if ( C <_ B , B , C ) ) -> A <_ if ( C <_ B , B , C ) ) )
10 3 9 mpan2d 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR ) -> ( A <_ B -> A <_ if ( C <_ B , B , C ) ) )
11 10 3impia 
 |-  ( ( ( B e. RR /\ C e. RR ) /\ A e. RR /\ A <_ B ) -> A <_ if ( C <_ B , B , C ) )