| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fldsdrgfldext.1 |
|- G = ( F |`s A ) |
| 2 |
|
fldsdrgfldext.2 |
|- ( ph -> F e. Field ) |
| 3 |
|
fldsdrgfldext.3 |
|- ( ph -> A e. ( SubDRing ` F ) ) |
| 4 |
|
fldsdrgfldext2.b |
|- ( ph -> B e. ( SubDRing ` G ) ) |
| 5 |
|
fldsdrgfldext2.h |
|- H = ( F |`s B ) |
| 6 |
|
eqid |
|- ( G |`s B ) = ( G |`s B ) |
| 7 |
|
fldsdrgfld |
|- ( ( F e. Field /\ A e. ( SubDRing ` F ) ) -> ( F |`s A ) e. Field ) |
| 8 |
2 3 7
|
syl2anc |
|- ( ph -> ( F |`s A ) e. Field ) |
| 9 |
1 8
|
eqeltrid |
|- ( ph -> G e. Field ) |
| 10 |
6 9 4
|
fldsdrgfldext |
|- ( ph -> G /FldExt ( G |`s B ) ) |
| 11 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
| 12 |
11
|
sdrgss |
|- ( B e. ( SubDRing ` G ) -> B C_ ( Base ` G ) ) |
| 13 |
4 12
|
syl |
|- ( ph -> B C_ ( Base ` G ) ) |
| 14 |
|
eqid |
|- ( Base ` F ) = ( Base ` F ) |
| 15 |
14
|
sdrgss |
|- ( A e. ( SubDRing ` F ) -> A C_ ( Base ` F ) ) |
| 16 |
1 14
|
ressbas2 |
|- ( A C_ ( Base ` F ) -> A = ( Base ` G ) ) |
| 17 |
3 15 16
|
3syl |
|- ( ph -> A = ( Base ` G ) ) |
| 18 |
13 17
|
sseqtrrd |
|- ( ph -> B C_ A ) |
| 19 |
|
ressabs |
|- ( ( A e. ( SubDRing ` F ) /\ B C_ A ) -> ( ( F |`s A ) |`s B ) = ( F |`s B ) ) |
| 20 |
3 18 19
|
syl2anc |
|- ( ph -> ( ( F |`s A ) |`s B ) = ( F |`s B ) ) |
| 21 |
1
|
oveq1i |
|- ( G |`s B ) = ( ( F |`s A ) |`s B ) |
| 22 |
20 21 5
|
3eqtr4g |
|- ( ph -> ( G |`s B ) = H ) |
| 23 |
10 22
|
breqtrd |
|- ( ph -> G /FldExt H ) |