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 ) ) )