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 ) ) ) |