Step |
Hyp |
Ref |
Expression |
1 |
|
fldextfld1 |
|- ( E /FldExt F -> E e. Field ) |
2 |
|
isfld |
|- ( E e. Field <-> ( E e. DivRing /\ E e. CRing ) ) |
3 |
2
|
simplbi |
|- ( E e. Field -> E e. DivRing ) |
4 |
1 3
|
syl |
|- ( E /FldExt F -> E e. DivRing ) |
5 |
|
fldextress |
|- ( E /FldExt F -> F = ( E |`s ( Base ` F ) ) ) |
6 |
|
fldextfld2 |
|- ( E /FldExt F -> F e. Field ) |
7 |
|
isfld |
|- ( F e. Field <-> ( F e. DivRing /\ F e. CRing ) ) |
8 |
7
|
simplbi |
|- ( F e. Field -> F e. DivRing ) |
9 |
6 8
|
syl |
|- ( E /FldExt F -> F e. DivRing ) |
10 |
5 9
|
eqeltrrd |
|- ( E /FldExt F -> ( E |`s ( Base ` F ) ) e. DivRing ) |
11 |
|
eqid |
|- ( Base ` F ) = ( Base ` F ) |
12 |
11
|
fldextsubrg |
|- ( E /FldExt F -> ( Base ` F ) e. ( SubRing ` E ) ) |
13 |
|
eqid |
|- ( ( subringAlg ` E ) ` ( Base ` F ) ) = ( ( subringAlg ` E ) ` ( Base ` F ) ) |
14 |
|
eqid |
|- ( E |`s ( Base ` F ) ) = ( E |`s ( Base ` F ) ) |
15 |
13 14
|
sralvec |
|- ( ( E e. DivRing /\ ( E |`s ( Base ` F ) ) e. DivRing /\ ( Base ` F ) e. ( SubRing ` E ) ) -> ( ( subringAlg ` E ) ` ( Base ` F ) ) e. LVec ) |
16 |
4 10 12 15
|
syl3anc |
|- ( E /FldExt F -> ( ( subringAlg ` E ) ` ( Base ` F ) ) e. LVec ) |
17 |
|
eqid |
|- ( Base ` E ) = ( Base ` E ) |
18 |
17
|
subrgss |
|- ( ( Base ` F ) e. ( SubRing ` E ) -> ( Base ` F ) C_ ( Base ` E ) ) |
19 |
12 18
|
syl |
|- ( E /FldExt F -> ( Base ` F ) C_ ( Base ` E ) ) |
20 |
13 17
|
sradrng |
|- ( ( E e. DivRing /\ ( Base ` F ) C_ ( Base ` E ) ) -> ( ( subringAlg ` E ) ` ( Base ` F ) ) e. DivRing ) |
21 |
4 19 20
|
syl2anc |
|- ( E /FldExt F -> ( ( subringAlg ` E ) ` ( Base ` F ) ) e. DivRing ) |
22 |
|
drngdimgt0 |
|- ( ( ( ( subringAlg ` E ) ` ( Base ` F ) ) e. LVec /\ ( ( subringAlg ` E ) ` ( Base ` F ) ) e. DivRing ) -> 0 < ( dim ` ( ( subringAlg ` E ) ` ( Base ` F ) ) ) ) |
23 |
16 21 22
|
syl2anc |
|- ( E /FldExt F -> 0 < ( dim ` ( ( subringAlg ` E ) ` ( Base ` F ) ) ) ) |
24 |
|
extdgval |
|- ( E /FldExt F -> ( E [:] F ) = ( dim ` ( ( subringAlg ` E ) ` ( Base ` F ) ) ) ) |
25 |
23 24
|
breqtrrd |
|- ( E /FldExt F -> 0 < ( E [:] F ) ) |