Step |
Hyp |
Ref |
Expression |
1 |
|
gsummonply1.p |
|- P = ( Poly1 ` R ) |
2 |
|
gsummonply1.b |
|- B = ( Base ` P ) |
3 |
|
gsummonply1.x |
|- X = ( var1 ` R ) |
4 |
|
gsummonply1.e |
|- .^ = ( .g ` ( mulGrp ` P ) ) |
5 |
|
gsummonply1.r |
|- ( ph -> R e. Ring ) |
6 |
|
gsummonply1.k |
|- K = ( Base ` R ) |
7 |
|
gsummonply1.m |
|- .* = ( .s ` P ) |
8 |
|
gsummonply1.0 |
|- .0. = ( 0g ` R ) |
9 |
|
gsummonply1.a |
|- ( ph -> A. k e. NN0 A e. K ) |
10 |
|
gsummonply1.f |
|- ( ph -> ( k e. NN0 |-> A ) finSupp .0. ) |
11 |
|
eqid |
|- ( 0g ` P ) = ( 0g ` P ) |
12 |
1
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
13 |
|
ringcmn |
|- ( P e. Ring -> P e. CMnd ) |
14 |
5 12 13
|
3syl |
|- ( ph -> P e. CMnd ) |
15 |
|
nn0ex |
|- NN0 e. _V |
16 |
15
|
a1i |
|- ( ph -> NN0 e. _V ) |
17 |
9
|
r19.21bi |
|- ( ( ph /\ k e. NN0 ) -> A e. K ) |
18 |
5
|
3ad2ant1 |
|- ( ( ph /\ k e. NN0 /\ A e. K ) -> R e. Ring ) |
19 |
|
simp3 |
|- ( ( ph /\ k e. NN0 /\ A e. K ) -> A e. K ) |
20 |
|
simp2 |
|- ( ( ph /\ k e. NN0 /\ A e. K ) -> k e. NN0 ) |
21 |
|
eqid |
|- ( mulGrp ` P ) = ( mulGrp ` P ) |
22 |
6 1 3 7 21 4 2
|
ply1tmcl |
|- ( ( R e. Ring /\ A e. K /\ k e. NN0 ) -> ( A .* ( k .^ X ) ) e. B ) |
23 |
18 19 20 22
|
syl3anc |
|- ( ( ph /\ k e. NN0 /\ A e. K ) -> ( A .* ( k .^ X ) ) e. B ) |
24 |
17 23
|
mpd3an3 |
|- ( ( ph /\ k e. NN0 ) -> ( A .* ( k .^ X ) ) e. B ) |
25 |
24
|
fmpttd |
|- ( ph -> ( k e. NN0 |-> ( A .* ( k .^ X ) ) ) : NN0 --> B ) |
26 |
1
|
ply1lmod |
|- ( R e. Ring -> P e. LMod ) |
27 |
5 26
|
syl |
|- ( ph -> P e. LMod ) |
28 |
1
|
ply1sca |
|- ( R e. Ring -> R = ( Scalar ` P ) ) |
29 |
5 28
|
syl |
|- ( ph -> R = ( Scalar ` P ) ) |
30 |
1 3 21 4 2
|
ply1moncl |
|- ( ( R e. Ring /\ k e. NN0 ) -> ( k .^ X ) e. B ) |
31 |
5 30
|
sylan |
|- ( ( ph /\ k e. NN0 ) -> ( k .^ X ) e. B ) |
32 |
16 27 29 2 17 31 11 8 7 10
|
mptscmfsupp0 |
|- ( ph -> ( k e. NN0 |-> ( A .* ( k .^ X ) ) ) finSupp ( 0g ` P ) ) |
33 |
2 11 14 16 25 32
|
gsumcl |
|- ( ph -> ( P gsum ( k e. NN0 |-> ( A .* ( k .^ X ) ) ) ) e. B ) |