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Metamath Proof Explorer

Theorem absdifle

Description: The absolute value of a difference and 'less than or equal to' relation. (Contributed by Paul Chapman, 18-Sep-2007)

Ref Expression
Assertion absdifle 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )

Proof

Step Hyp Ref Expression
1 resubcl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
2 absle 
 |-  ( ( ( A - B ) e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( -u C <_ ( A - B ) /\ ( A - B ) <_ C ) ) )
3 1 2 stoic3 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( -u C <_ ( A - B ) /\ ( A - B ) <_ C ) ) )
4 renegcl 
 |-  ( C e. RR -> -u C e. RR )
5 leaddsub2 
 |-  ( ( B e. RR /\ -u C e. RR /\ A e. RR ) -> ( ( B + -u C ) <_ A <-> -u C <_ ( A - B ) ) )
6 4 5 syl3an2 
 |-  ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( ( B + -u C ) <_ A <-> -u C <_ ( A - B ) ) )
7 6 3comr 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + -u C ) <_ A <-> -u C <_ ( A - B ) ) )
8 recn 
 |-  ( B e. RR -> B e. CC )
9 recn 
 |-  ( C e. RR -> C e. CC )
10 negsub 
 |-  ( ( B e. CC /\ C e. CC ) -> ( B + -u C ) = ( B - C ) )
11 8 9 10 syl2an 
 |-  ( ( B e. RR /\ C e. RR ) -> ( B + -u C ) = ( B - C ) )
12 11 3adant1 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + -u C ) = ( B - C ) )
13 12 breq1d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + -u C ) <_ A <-> ( B - C ) <_ A ) )
14 7 13 bitr3d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -u C <_ ( A - B ) <-> ( B - C ) <_ A ) )
15 lesubadd2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> A <_ ( B + C ) ) )
16 14 15 anbi12d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( -u C <_ ( A - B ) /\ ( A - B ) <_ C ) <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )
17 3 16 bitrd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( A - B ) ) <_ C <-> ( ( B - C ) <_ A /\ A <_ ( B + C ) ) ) )