Step |
Hyp |
Ref |
Expression |
1 |
|
evl1gsumadd.q |
|- Q = ( eval1 ` R ) |
2 |
|
evl1gsumadd.k |
|- K = ( Base ` R ) |
3 |
|
evl1gsumadd.w |
|- W = ( Poly1 ` R ) |
4 |
|
evl1gsumadd.p |
|- P = ( R ^s K ) |
5 |
|
evl1gsumadd.b |
|- B = ( Base ` W ) |
6 |
|
evl1gsumadd.r |
|- ( ph -> R e. CRing ) |
7 |
|
evl1gsumadd.y |
|- ( ( ph /\ x e. N ) -> Y e. B ) |
8 |
|
evl1gsumadd.n |
|- ( ph -> N C_ NN0 ) |
9 |
|
evl1gsumadd.0 |
|- .0. = ( 0g ` W ) |
10 |
|
evl1gsumadd.f |
|- ( ph -> ( x e. N |-> Y ) finSupp .0. ) |
11 |
1 2
|
evl1fval1 |
|- Q = ( R evalSub1 K ) |
12 |
11
|
a1i |
|- ( ph -> Q = ( R evalSub1 K ) ) |
13 |
12
|
fveq1d |
|- ( ph -> ( Q ` ( W gsum ( x e. N |-> Y ) ) ) = ( ( R evalSub1 K ) ` ( W gsum ( x e. N |-> Y ) ) ) ) |
14 |
2
|
ressid |
|- ( R e. CRing -> ( R |`s K ) = R ) |
15 |
6 14
|
syl |
|- ( ph -> ( R |`s K ) = R ) |
16 |
15
|
eqcomd |
|- ( ph -> R = ( R |`s K ) ) |
17 |
16
|
fveq2d |
|- ( ph -> ( Poly1 ` R ) = ( Poly1 ` ( R |`s K ) ) ) |
18 |
3 17
|
syl5eq |
|- ( ph -> W = ( Poly1 ` ( R |`s K ) ) ) |
19 |
18
|
fvoveq1d |
|- ( ph -> ( ( R evalSub1 K ) ` ( W gsum ( x e. N |-> Y ) ) ) = ( ( R evalSub1 K ) ` ( ( Poly1 ` ( R |`s K ) ) gsum ( x e. N |-> Y ) ) ) ) |
20 |
|
eqid |
|- ( R evalSub1 K ) = ( R evalSub1 K ) |
21 |
|
eqid |
|- ( Poly1 ` ( R |`s K ) ) = ( Poly1 ` ( R |`s K ) ) |
22 |
|
eqid |
|- ( 0g ` ( Poly1 ` ( R |`s K ) ) ) = ( 0g ` ( Poly1 ` ( R |`s K ) ) ) |
23 |
|
eqid |
|- ( R |`s K ) = ( R |`s K ) |
24 |
|
eqid |
|- ( Base ` ( Poly1 ` ( R |`s K ) ) ) = ( Base ` ( Poly1 ` ( R |`s K ) ) ) |
25 |
|
crngring |
|- ( R e. CRing -> R e. Ring ) |
26 |
2
|
subrgid |
|- ( R e. Ring -> K e. ( SubRing ` R ) ) |
27 |
6 25 26
|
3syl |
|- ( ph -> K e. ( SubRing ` R ) ) |
28 |
18
|
adantr |
|- ( ( ph /\ x e. N ) -> W = ( Poly1 ` ( R |`s K ) ) ) |
29 |
28
|
fveq2d |
|- ( ( ph /\ x e. N ) -> ( Base ` W ) = ( Base ` ( Poly1 ` ( R |`s K ) ) ) ) |
30 |
5 29
|
syl5eq |
|- ( ( ph /\ x e. N ) -> B = ( Base ` ( Poly1 ` ( R |`s K ) ) ) ) |
31 |
7 30
|
eleqtrd |
|- ( ( ph /\ x e. N ) -> Y e. ( Base ` ( Poly1 ` ( R |`s K ) ) ) ) |
32 |
18
|
eqcomd |
|- ( ph -> ( Poly1 ` ( R |`s K ) ) = W ) |
33 |
32
|
fveq2d |
|- ( ph -> ( 0g ` ( Poly1 ` ( R |`s K ) ) ) = ( 0g ` W ) ) |
34 |
33 9
|
eqtr4di |
|- ( ph -> ( 0g ` ( Poly1 ` ( R |`s K ) ) ) = .0. ) |
35 |
10 34
|
breqtrrd |
|- ( ph -> ( x e. N |-> Y ) finSupp ( 0g ` ( Poly1 ` ( R |`s K ) ) ) ) |
36 |
20 2 21 22 23 4 24 6 27 31 8 35
|
evls1gsumadd |
|- ( ph -> ( ( R evalSub1 K ) ` ( ( Poly1 ` ( R |`s K ) ) gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( ( R evalSub1 K ) ` Y ) ) ) ) |
37 |
19 36
|
eqtrd |
|- ( ph -> ( ( R evalSub1 K ) ` ( W gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( ( R evalSub1 K ) ` Y ) ) ) ) |
38 |
12
|
fveq1d |
|- ( ph -> ( Q ` Y ) = ( ( R evalSub1 K ) ` Y ) ) |
39 |
38
|
eqcomd |
|- ( ph -> ( ( R evalSub1 K ) ` Y ) = ( Q ` Y ) ) |
40 |
39
|
mpteq2dv |
|- ( ph -> ( x e. N |-> ( ( R evalSub1 K ) ` Y ) ) = ( x e. N |-> ( Q ` Y ) ) ) |
41 |
40
|
oveq2d |
|- ( ph -> ( P gsum ( x e. N |-> ( ( R evalSub1 K ) ` Y ) ) ) = ( P gsum ( x e. N |-> ( Q ` Y ) ) ) ) |
42 |
13 37 41
|
3eqtrd |
|- ( ph -> ( Q ` ( W gsum ( x e. N |-> Y ) ) ) = ( P gsum ( x e. N |-> ( Q ` Y ) ) ) ) |