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Metamath Proof Explorer

Theorem avgle

Description: The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005) (Revised by Mario Carneiro, 28-May-2014)

Ref Expression
Assertion avgle 
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) )

Proof

Step Hyp Ref Expression
1 letric 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B \/ B <_ A ) )
2 1 orcomd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B <_ A \/ A <_ B ) )
3 avgle2 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B <_ A <-> ( ( B + A ) / 2 ) <_ A ) )
4 3 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> ( ( B + A ) / 2 ) <_ A ) )
5 recn 
 |-  ( A e. RR -> A e. CC )
6 recn 
 |-  ( B e. RR -> B e. CC )
7 addcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
8 5 6 7 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
9 8 oveq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) / 2 ) = ( ( B + A ) / 2 ) )
10 9 breq1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) / 2 ) <_ A <-> ( ( B + A ) / 2 ) <_ A ) )
11 4 10 bitr4d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B <_ A <-> ( ( A + B ) / 2 ) <_ A ) )
12 avgle2 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( ( A + B ) / 2 ) <_ B ) )
13 11 12 orbi12d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( B <_ A \/ A <_ B ) <-> ( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) ) )
14 2 13 mpbid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) / 2 ) <_ A \/ ( ( A + B ) / 2 ) <_ B ) )