Step |
Hyp |
Ref |
Expression |
1 |
|
cjf |
|- * : CC --> CC |
2 |
|
cnex |
|- CC e. _V |
3 |
|
fex2 |
|- ( ( * : CC --> CC /\ CC e. _V /\ CC e. _V ) -> * e. _V ) |
4 |
1 2 2 3
|
mp3an |
|- * e. _V |
5 |
|
cnfldstr |
|- CCfld Struct <. 1 , ; 1 3 >. |
6 |
|
starvid |
|- *r = Slot ( *r ` ndx ) |
7 |
|
ssun2 |
|- { <. ( *r ` ndx ) , * >. } C_ ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) |
8 |
|
ssun1 |
|- ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) C_ ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. ( TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , ( metUnif ` ( abs o. - ) ) >. } ) ) |
9 |
|
df-cnfld |
|- CCfld = ( ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) u. ( { <. ( TopSet ` ndx ) , ( MetOpen ` ( abs o. - ) ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( abs o. - ) >. } u. { <. ( UnifSet ` ndx ) , ( metUnif ` ( abs o. - ) ) >. } ) ) |
10 |
8 9
|
sseqtrri |
|- ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } ) C_ CCfld |
11 |
7 10
|
sstri |
|- { <. ( *r ` ndx ) , * >. } C_ CCfld |
12 |
5 6 11
|
strfv |
|- ( * e. _V -> * = ( *r ` CCfld ) ) |
13 |
4 12
|
ax-mp |
|- * = ( *r ` CCfld ) |