Step |
Hyp |
Ref |
Expression |
1 |
|
ltaddsub |
|- ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) ) |
2 |
1
|
3com13 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) ) |
3 |
2
|
3expa |
|- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( ( C + B ) < A <-> C < ( A - B ) ) ) |
4 |
3
|
adantrr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( C + B ) < A <-> C < ( A - B ) ) ) |
5 |
|
ltsubadd |
|- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( ( A - B ) < D <-> A < ( D + B ) ) ) |
6 |
5
|
bicomd |
|- ( ( A e. RR /\ B e. RR /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) ) |
7 |
6
|
3expa |
|- ( ( ( A e. RR /\ B e. RR ) /\ D e. RR ) -> ( A < ( D + B ) <-> ( A - B ) < D ) ) |
8 |
7
|
adantrl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < ( D + B ) <-> ( A - B ) < D ) ) |
9 |
4 8
|
anbi12d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( C + B ) < A /\ A < ( D + B ) ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) ) |
10 |
|
readdcl |
|- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR ) |
11 |
10
|
rexrd |
|- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR* ) |
12 |
11
|
ad2ant2rl |
|- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( C + B ) e. RR* ) |
13 |
|
readdcl |
|- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR ) |
14 |
13
|
rexrd |
|- ( ( D e. RR /\ B e. RR ) -> ( D + B ) e. RR* ) |
15 |
14
|
ad2ant2l |
|- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( D + B ) e. RR* ) |
16 |
|
rexr |
|- ( A e. RR -> A e. RR* ) |
17 |
16
|
ad2antrl |
|- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> A e. RR* ) |
18 |
|
elioo5 |
|- ( ( ( C + B ) e. RR* /\ ( D + B ) e. RR* /\ A e. RR* ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) ) |
19 |
12 15 17 18
|
syl3anc |
|- ( ( ( C e. RR /\ D e. RR ) /\ ( A e. RR /\ B e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) ) |
20 |
19
|
ancoms |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A e. ( ( C + B ) (,) ( D + B ) ) <-> ( ( C + B ) < A /\ A < ( D + B ) ) ) ) |
21 |
|
rexr |
|- ( C e. RR -> C e. RR* ) |
22 |
21
|
ad2antrl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR* ) |
23 |
|
rexr |
|- ( D e. RR -> D e. RR* ) |
24 |
23
|
ad2antll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR* ) |
25 |
|
resubcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR ) |
26 |
25
|
rexrd |
|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR* ) |
27 |
26
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR* ) |
28 |
|
elioo5 |
|- ( ( C e. RR* /\ D e. RR* /\ ( A - B ) e. RR* ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) ) |
29 |
22 24 27 28
|
syl3anc |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> ( C < ( A - B ) /\ ( A - B ) < D ) ) ) |
30 |
9 20 29
|
3bitr4rd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A - B ) e. ( C (,) D ) <-> A e. ( ( C + B ) (,) ( D + B ) ) ) ) |