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Metamath Proof Explorer

Theorem absdiflt

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

Ref Expression
Assertion absdiflt 
|- ( ( 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 abslt 
 |-  ( ( ( 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 ltaddsub2 
 |-  ( ( 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 ltsubadd2 
 |-  ( ( 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 ) ) ) )