Step |
Hyp |
Ref |
Expression |
1 |
|
algextdeg.k |
|- K = ( E |`s F ) |
2 |
|
algextdeg.l |
|- L = ( E |`s ( E fldGen ( F u. { A } ) ) ) |
3 |
|
algextdeg.d |
|- D = ( deg1 ` E ) |
4 |
|
algextdeg.m |
|- M = ( E minPoly F ) |
5 |
|
algextdeg.f |
|- ( ph -> E e. Field ) |
6 |
|
algextdeg.e |
|- ( ph -> F e. ( SubDRing ` E ) ) |
7 |
|
algextdeg.a |
|- ( ph -> A e. ( E IntgRing F ) ) |
8 |
|
algextdeglem.o |
|- O = ( E evalSub1 F ) |
9 |
|
algextdeglem.y |
|- P = ( Poly1 ` K ) |
10 |
|
algextdeglem.u |
|- U = ( Base ` P ) |
11 |
|
algextdeglem.g |
|- G = ( p e. U |-> ( ( O ` p ) ` A ) ) |
12 |
|
algextdeglem.n |
|- N = ( x e. U |-> [ x ] ( P ~QG Z ) ) |
13 |
|
algextdeglem.z |
|- Z = ( `' G " { ( 0g ` L ) } ) |
14 |
|
algextdeglem.q |
|- Q = ( P /s ( P ~QG Z ) ) |
15 |
|
algextdeglem.j |
|- J = ( p e. ( Base ` Q ) |-> U. ( G " p ) ) |
16 |
1
|
fveq2i |
|- ( Poly1 ` K ) = ( Poly1 ` ( E |`s F ) ) |
17 |
9 16
|
eqtri |
|- P = ( Poly1 ` ( E |`s F ) ) |
18 |
|
issdrg |
|- ( F e. ( SubDRing ` E ) <-> ( E e. DivRing /\ F e. ( SubRing ` E ) /\ ( E |`s F ) e. DivRing ) ) |
19 |
6 18
|
sylib |
|- ( ph -> ( E e. DivRing /\ F e. ( SubRing ` E ) /\ ( E |`s F ) e. DivRing ) ) |
20 |
19
|
simp3d |
|- ( ph -> ( E |`s F ) e. DivRing ) |
21 |
17 20
|
ply1lvec |
|- ( ph -> P e. LVec ) |
22 |
|
eqidd |
|- ( ph -> ( ( subringAlg ` L ) ` F ) = ( ( subringAlg ` L ) ` F ) ) |
23 |
|
eqidd |
|- ( ph -> ( 0g ` L ) = ( 0g ` L ) ) |
24 |
|
eqid |
|- ( Base ` E ) = ( Base ` E ) |
25 |
5
|
flddrngd |
|- ( ph -> E e. DivRing ) |
26 |
19
|
simp2d |
|- ( ph -> F e. ( SubRing ` E ) ) |
27 |
|
subrgsubg |
|- ( F e. ( SubRing ` E ) -> F e. ( SubGrp ` E ) ) |
28 |
24
|
subgss |
|- ( F e. ( SubGrp ` E ) -> F C_ ( Base ` E ) ) |
29 |
26 27 28
|
3syl |
|- ( ph -> F C_ ( Base ` E ) ) |
30 |
|
eqid |
|- ( 0g ` E ) = ( 0g ` E ) |
31 |
5
|
fldcrngd |
|- ( ph -> E e. CRing ) |
32 |
8 1 24 30 31 26
|
irngssv |
|- ( ph -> ( E IntgRing F ) C_ ( Base ` E ) ) |
33 |
32 7
|
sseldd |
|- ( ph -> A e. ( Base ` E ) ) |
34 |
33
|
snssd |
|- ( ph -> { A } C_ ( Base ` E ) ) |
35 |
29 34
|
unssd |
|- ( ph -> ( F u. { A } ) C_ ( Base ` E ) ) |
36 |
24 25 35
|
fldgenssid |
|- ( ph -> ( F u. { A } ) C_ ( E fldGen ( F u. { A } ) ) ) |
37 |
36
|
unssad |
|- ( ph -> F C_ ( E fldGen ( F u. { A } ) ) ) |
38 |
24 25 35
|
fldgenssv |
|- ( ph -> ( E fldGen ( F u. { A } ) ) C_ ( Base ` E ) ) |
39 |
2 24
|
ressbas2 |
|- ( ( E fldGen ( F u. { A } ) ) C_ ( Base ` E ) -> ( E fldGen ( F u. { A } ) ) = ( Base ` L ) ) |
40 |
38 39
|
syl |
|- ( ph -> ( E fldGen ( F u. { A } ) ) = ( Base ` L ) ) |
41 |
37 40
|
sseqtrd |
|- ( ph -> F C_ ( Base ` L ) ) |
42 |
22 23 41
|
sralmod0 |
|- ( ph -> ( 0g ` L ) = ( 0g ` ( ( subringAlg ` L ) ` F ) ) ) |
43 |
42
|
sneqd |
|- ( ph -> { ( 0g ` L ) } = { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) |
44 |
43
|
imaeq2d |
|- ( ph -> ( `' G " { ( 0g ` L ) } ) = ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) ) |
45 |
13 44
|
eqtrid |
|- ( ph -> Z = ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) ) |
46 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
algextdeglem2 |
|- ( ph -> G e. ( P LMHom ( ( subringAlg ` L ) ` F ) ) ) |
47 |
|
eqid |
|- ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) = ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) |
48 |
|
eqid |
|- ( 0g ` ( ( subringAlg ` L ) ` F ) ) = ( 0g ` ( ( subringAlg ` L ) ` F ) ) |
49 |
|
eqid |
|- ( LSubSp ` P ) = ( LSubSp ` P ) |
50 |
47 48 49
|
lmhmkerlss |
|- ( G e. ( P LMHom ( ( subringAlg ` L ) ` F ) ) -> ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) e. ( LSubSp ` P ) ) |
51 |
46 50
|
syl |
|- ( ph -> ( `' G " { ( 0g ` ( ( subringAlg ` L ) ` F ) ) } ) e. ( LSubSp ` P ) ) |
52 |
45 51
|
eqeltrd |
|- ( ph -> Z e. ( LSubSp ` P ) ) |
53 |
14 21 52
|
quslvec |
|- ( ph -> Q e. LVec ) |