| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fldextsdrg.1 |
|- B = ( Base ` F ) |
| 2 |
|
fldextsdrg.2 |
|- ( ph -> E /FldExt F ) |
| 3 |
|
fldextfld1 |
|- ( E /FldExt F -> E e. Field ) |
| 4 |
2 3
|
syl |
|- ( ph -> E e. Field ) |
| 5 |
4
|
flddrngd |
|- ( ph -> E e. DivRing ) |
| 6 |
1
|
fldextsubrg |
|- ( E /FldExt F -> B e. ( SubRing ` E ) ) |
| 7 |
2 6
|
syl |
|- ( ph -> B e. ( SubRing ` E ) ) |
| 8 |
|
fldextress |
|- ( E /FldExt F -> F = ( E |`s ( Base ` F ) ) ) |
| 9 |
2 8
|
syl |
|- ( ph -> F = ( E |`s ( Base ` F ) ) ) |
| 10 |
1
|
oveq2i |
|- ( E |`s B ) = ( E |`s ( Base ` F ) ) |
| 11 |
9 10
|
eqtr4di |
|- ( ph -> F = ( E |`s B ) ) |
| 12 |
|
fldextfld2 |
|- ( E /FldExt F -> F e. Field ) |
| 13 |
2 12
|
syl |
|- ( ph -> F e. Field ) |
| 14 |
11 13
|
eqeltrrd |
|- ( ph -> ( E |`s B ) e. Field ) |
| 15 |
14
|
flddrngd |
|- ( ph -> ( E |`s B ) e. DivRing ) |
| 16 |
|
issdrg |
|- ( B e. ( SubDRing ` E ) <-> ( E e. DivRing /\ B e. ( SubRing ` E ) /\ ( E |`s B ) e. DivRing ) ) |
| 17 |
5 7 15 16
|
syl3anbrc |
|- ( ph -> B e. ( SubDRing ` E ) ) |