Step |
Hyp |
Ref |
Expression |
1 |
|
remet.1 |
|- D = ( ( abs o. - ) |` ( RR X. RR ) ) |
2 |
1
|
ioo2bl |
|- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) = ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) ) |
3 |
1
|
rexmet |
|- D e. ( *Met ` RR ) |
4 |
|
readdcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) |
5 |
4
|
rehalfcld |
|- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) / 2 ) e. RR ) |
6 |
|
resubcl |
|- ( ( B e. RR /\ A e. RR ) -> ( B - A ) e. RR ) |
7 |
6
|
ancoms |
|- ( ( A e. RR /\ B e. RR ) -> ( B - A ) e. RR ) |
8 |
7
|
rehalfcld |
|- ( ( A e. RR /\ B e. RR ) -> ( ( B - A ) / 2 ) e. RR ) |
9 |
8
|
rexrd |
|- ( ( A e. RR /\ B e. RR ) -> ( ( B - A ) / 2 ) e. RR* ) |
10 |
|
blelrn |
|- ( ( D e. ( *Met ` RR ) /\ ( ( A + B ) / 2 ) e. RR /\ ( ( B - A ) / 2 ) e. RR* ) -> ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) e. ran ( ball ` D ) ) |
11 |
3 5 9 10
|
mp3an2i |
|- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) / 2 ) ( ball ` D ) ( ( B - A ) / 2 ) ) e. ran ( ball ` D ) ) |
12 |
2 11
|
eqeltrd |
|- ( ( A e. RR /\ B e. RR ) -> ( A (,) B ) e. ran ( ball ` D ) ) |