Step |
Hyp |
Ref |
Expression |
1 |
|
deg1pw.d |
|- D = ( deg1 ` R ) |
2 |
|
deg1pw.p |
|- P = ( Poly1 ` R ) |
3 |
|
deg1pw.x |
|- X = ( var1 ` R ) |
4 |
|
deg1pw.n |
|- N = ( mulGrp ` P ) |
5 |
|
deg1pw.e |
|- .^ = ( .g ` N ) |
6 |
2
|
ply1sca |
|- ( R e. NzRing -> R = ( Scalar ` P ) ) |
7 |
6
|
adantr |
|- ( ( R e. NzRing /\ F e. NN0 ) -> R = ( Scalar ` P ) ) |
8 |
7
|
fveq2d |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) = ( 1r ` ( Scalar ` P ) ) ) |
9 |
8
|
oveq1d |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) = ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) ) |
10 |
|
nzrring |
|- ( R e. NzRing -> R e. Ring ) |
11 |
10
|
adantr |
|- ( ( R e. NzRing /\ F e. NN0 ) -> R e. Ring ) |
12 |
2
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
13 |
11 12
|
syl |
|- ( ( R e. NzRing /\ F e. NN0 ) -> P e. LMod ) |
14 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
15 |
4 14
|
mgpbas |
|- ( Base ` P ) = ( Base ` N ) |
16 |
2
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
17 |
4
|
ringmgp |
|- ( P e. Ring -> N e. Mnd ) |
18 |
11 16 17
|
3syl |
|- ( ( R e. NzRing /\ F e. NN0 ) -> N e. Mnd ) |
19 |
|
simpr |
|- ( ( R e. NzRing /\ F e. NN0 ) -> F e. NN0 ) |
20 |
3 2 14
|
vr1cl |
|- ( R e. Ring -> X e. ( Base ` P ) ) |
21 |
11 20
|
syl |
|- ( ( R e. NzRing /\ F e. NN0 ) -> X e. ( Base ` P ) ) |
22 |
15 5 18 19 21
|
mulgnn0cld |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( F .^ X ) e. ( Base ` P ) ) |
23 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
24 |
|
eqid |
|- ( .s ` P ) = ( .s ` P ) |
25 |
|
eqid |
|- ( 1r ` ( Scalar ` P ) ) = ( 1r ` ( Scalar ` P ) ) |
26 |
14 23 24 25
|
lmodvs1 |
|- ( ( P e. LMod /\ ( F .^ X ) e. ( Base ` P ) ) -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) ) |
27 |
13 22 26
|
syl2anc |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` ( Scalar ` P ) ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) ) |
28 |
9 27
|
eqtrd |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) = ( F .^ X ) ) |
29 |
28
|
fveq2d |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = ( D ` ( F .^ X ) ) ) |
30 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
31 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
32 |
30 31
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
33 |
11 32
|
syl |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) e. ( Base ` R ) ) |
34 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
35 |
31 34
|
nzrnz |
|- ( R e. NzRing -> ( 1r ` R ) =/= ( 0g ` R ) ) |
36 |
35
|
adantr |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( 1r ` R ) =/= ( 0g ` R ) ) |
37 |
1 30 2 3 24 4 5 34
|
deg1tm |
|- ( ( R e. Ring /\ ( ( 1r ` R ) e. ( Base ` R ) /\ ( 1r ` R ) =/= ( 0g ` R ) ) /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = F ) |
38 |
11 33 36 19 37
|
syl121anc |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( ( 1r ` R ) ( .s ` P ) ( F .^ X ) ) ) = F ) |
39 |
29 38
|
eqtr3d |
|- ( ( R e. NzRing /\ F e. NN0 ) -> ( D ` ( F .^ X ) ) = F ) |