Step |
Hyp |
Ref |
Expression |
1 |
|
resubcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR ) |
2 |
1
|
3adant3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - B ) e. RR ) |
3 |
2
|
rexrd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - B ) e. RR* ) |
4 |
|
readdcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) |
5 |
4
|
3adant3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + B ) e. RR ) |
6 |
5
|
rexrd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + B ) e. RR* ) |
7 |
|
rexr |
|- ( C e. RR -> C e. RR* ) |
8 |
7
|
3ad2ant3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR* ) |
9 |
|
elicc4 |
|- ( ( ( A - B ) e. RR* /\ ( A + B ) e. RR* /\ C e. RR* ) -> ( C e. ( ( A - B ) [,] ( A + B ) ) <-> ( ( A - B ) <_ C /\ C <_ ( A + B ) ) ) ) |
10 |
3 6 8 9
|
syl3anc |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( ( A - B ) [,] ( A + B ) ) <-> ( ( A - B ) <_ C /\ C <_ ( A + B ) ) ) ) |
11 |
|
absdifle |
|- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( abs ` ( C - A ) ) <_ B <-> ( ( A - B ) <_ C /\ C <_ ( A + B ) ) ) ) |
12 |
11
|
3coml |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( abs ` ( C - A ) ) <_ B <-> ( ( A - B ) <_ C /\ C <_ ( A + B ) ) ) ) |
13 |
10 12
|
bitr4d |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C e. ( ( A - B ) [,] ( A + B ) ) <-> ( abs ` ( C - A ) ) <_ B ) ) |