| Step |
Hyp |
Ref |
Expression |
| 1 |
|
absvalsqi.1 |
|- A e. CC |
| 2 |
|
abssub.2 |
|- B e. CC |
| 3 |
|
abs3dif.3 |
|- C e. CC |
| 4 |
|
abs3lem.4 |
|- D e. RR |
| 5 |
1 2 3
|
abs3difi |
|- ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) |
| 6 |
1 3
|
subcli |
|- ( A - C ) e. CC |
| 7 |
6
|
abscli |
|- ( abs ` ( A - C ) ) e. RR |
| 8 |
3 2
|
subcli |
|- ( C - B ) e. CC |
| 9 |
8
|
abscli |
|- ( abs ` ( C - B ) ) e. RR |
| 10 |
4
|
rehalfcli |
|- ( D / 2 ) e. RR |
| 11 |
7 9 10 10
|
lt2addi |
|- ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) < ( ( D / 2 ) + ( D / 2 ) ) ) |
| 12 |
1 2
|
subcli |
|- ( A - B ) e. CC |
| 13 |
12
|
abscli |
|- ( abs ` ( A - B ) ) e. RR |
| 14 |
7 9
|
readdcli |
|- ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) e. RR |
| 15 |
10 10
|
readdcli |
|- ( ( D / 2 ) + ( D / 2 ) ) e. RR |
| 16 |
13 14 15
|
lelttri |
|- ( ( ( abs ` ( A - B ) ) <_ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) /\ ( ( abs ` ( A - C ) ) + ( abs ` ( C - B ) ) ) < ( ( D / 2 ) + ( D / 2 ) ) ) -> ( abs ` ( A - B ) ) < ( ( D / 2 ) + ( D / 2 ) ) ) |
| 17 |
5 11 16
|
sylancr |
|- ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < ( ( D / 2 ) + ( D / 2 ) ) ) |
| 18 |
10
|
recni |
|- ( D / 2 ) e. CC |
| 19 |
18
|
2timesi |
|- ( 2 x. ( D / 2 ) ) = ( ( D / 2 ) + ( D / 2 ) ) |
| 20 |
4
|
recni |
|- D e. CC |
| 21 |
|
2cn |
|- 2 e. CC |
| 22 |
|
2ne0 |
|- 2 =/= 0 |
| 23 |
20 21 22
|
divcan2i |
|- ( 2 x. ( D / 2 ) ) = D |
| 24 |
19 23
|
eqtr3i |
|- ( ( D / 2 ) + ( D / 2 ) ) = D |
| 25 |
17 24
|
breqtrdi |
|- ( ( ( abs ` ( A - C ) ) < ( D / 2 ) /\ ( abs ` ( C - B ) ) < ( D / 2 ) ) -> ( abs ` ( A - B ) ) < D ) |