Metamath Proof Explorer

Theorem abslt

Description: Absolute value and 'less than' relation. (Contributed by NM, 6-Apr-2005) (Revised by Mario Carneiro, 29-May-2016)

Ref Expression
Assertion abslt 
|- ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) )

Proof

Step Hyp Ref Expression
1 simpll 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. RR )
2 1 renegcld 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A e. RR )
3 1 recnd 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A e. CC )
4 abscl 
 |-  ( A e. CC -> ( abs ` A ) e. RR )
5 3 4 syl 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) e. RR )
6 simplr 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> B e. RR )
7 leabs 
 |-  ( -u A e. RR -> -u A <_ ( abs ` -u A ) )
8 2 7 syl 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A <_ ( abs ` -u A ) )
9 absneg 
 |-  ( A e. CC -> ( abs ` -u A ) = ( abs ` A ) )
10 3 9 syl 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` -u A ) = ( abs ` A ) )
11 8 10 breqtrd 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A <_ ( abs ` A ) )
12 simpr 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( abs ` A ) < B )
13 2 5 6 11 12 lelttrd 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> -u A < B )
14 leabs 
 |-  ( A e. RR -> A <_ ( abs ` A ) )
15 14 ad2antrr 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A <_ ( abs ` A ) )
16 1 5 6 15 12 lelttrd 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> A < B )
17 13 16 jca 
 |-  ( ( ( A e. RR /\ B e. RR ) /\ ( abs ` A ) < B ) -> ( -u A < B /\ A < B ) )
18 17 ex 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B -> ( -u A < B /\ A < B ) ) )
19 absor 
 |-  ( A e. RR -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
20 19 adantr 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) )
21 breq1 
 |-  ( ( abs ` A ) = A -> ( ( abs ` A ) < B <-> A < B ) )
22 21 biimprd 
 |-  ( ( abs ` A ) = A -> ( A < B -> ( abs ` A ) < B ) )
23 breq1 
 |-  ( ( abs ` A ) = -u A -> ( ( abs ` A ) < B <-> -u A < B ) )
24 23 biimprd 
 |-  ( ( abs ` A ) = -u A -> ( -u A < B -> ( abs ` A ) < B ) )
25 22 24 jaoa 
 |-  ( ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) -> ( ( A < B /\ -u A < B ) -> ( abs ` A ) < B ) )
26 25 ancomsd 
 |-  ( ( ( abs ` A ) = A \/ ( abs ` A ) = -u A ) -> ( ( -u A < B /\ A < B ) -> ( abs ` A ) < B ) )
27 20 26 syl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( -u A < B /\ A < B ) -> ( abs ` A ) < B ) )
28 18 27 impbid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u A < B /\ A < B ) ) )
29 ltnegcon1 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -u A < B <-> -u B < A ) )
30 29 anbi1d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( -u A < B /\ A < B ) <-> ( -u B < A /\ A < B ) ) )
31 28 30 bitrd 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( abs ` A ) < B <-> ( -u B < A /\ A < B ) ) )