Step |
Hyp |
Ref |
Expression |
1 |
|
ply1coefsupp.p |
|- P = ( Poly1 ` R ) |
2 |
|
ply1coefsupp.x |
|- X = ( var1 ` R ) |
3 |
|
ply1coefsupp.b |
|- B = ( Base ` P ) |
4 |
|
ply1coefsupp.n |
|- .x. = ( .s ` P ) |
5 |
|
ply1coefsupp.m |
|- M = ( mulGrp ` P ) |
6 |
|
ply1coefsupp.e |
|- .^ = ( .g ` M ) |
7 |
|
ply1coefsupp.a |
|- A = ( coe1 ` K ) |
8 |
|
eqid |
|- ( Scalar ` P ) = ( Scalar ` P ) |
9 |
1
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
10 |
9
|
adantr |
|- ( ( R e. Ring /\ K e. B ) -> P e. LMod ) |
11 |
|
nn0ex |
|- NN0 e. _V |
12 |
11
|
a1i |
|- ( ( R e. Ring /\ K e. B ) -> NN0 e. _V ) |
13 |
1
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
14 |
5
|
ringmgp |
|- ( P e. Ring -> M e. Mnd ) |
15 |
13 14
|
syl |
|- ( R e. Ring -> M e. Mnd ) |
16 |
15
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> M e. Mnd ) |
17 |
|
simpr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> k e. NN0 ) |
18 |
2 1 3
|
vr1cl |
|- ( R e. Ring -> X e. B ) |
19 |
18
|
ad2antrr |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> X e. B ) |
20 |
5 3
|
mgpbas |
|- B = ( Base ` M ) |
21 |
20 6
|
mulgnn0cl |
|- ( ( M e. Mnd /\ k e. NN0 /\ X e. B ) -> ( k .^ X ) e. B ) |
22 |
16 17 19 21
|
syl3anc |
|- ( ( ( R e. Ring /\ K e. B ) /\ k e. NN0 ) -> ( k .^ X ) e. B ) |
23 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
24 |
7 3 1 23
|
coe1f |
|- ( K e. B -> A : NN0 --> ( Base ` R ) ) |
25 |
24
|
adantl |
|- ( ( R e. Ring /\ K e. B ) -> A : NN0 --> ( Base ` R ) ) |
26 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
27 |
7 3 1 26
|
coe1sfi |
|- ( K e. B -> A finSupp ( 0g ` R ) ) |
28 |
27
|
adantl |
|- ( ( R e. Ring /\ K e. B ) -> A finSupp ( 0g ` R ) ) |
29 |
1
|
ply1sca |
|- ( R e. Ring -> R = ( Scalar ` P ) ) |
30 |
29
|
eqcomd |
|- ( R e. Ring -> ( Scalar ` P ) = R ) |
31 |
30
|
adantr |
|- ( ( R e. Ring /\ K e. B ) -> ( Scalar ` P ) = R ) |
32 |
31
|
fveq2d |
|- ( ( R e. Ring /\ K e. B ) -> ( 0g ` ( Scalar ` P ) ) = ( 0g ` R ) ) |
33 |
28 32
|
breqtrrd |
|- ( ( R e. Ring /\ K e. B ) -> A finSupp ( 0g ` ( Scalar ` P ) ) ) |
34 |
3 8 4 10 12 22 25 33
|
mptscmfsuppd |
|- ( ( R e. Ring /\ K e. B ) -> ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ X ) ) ) finSupp ( 0g ` P ) ) |