| Step |
Hyp |
Ref |
Expression |
| 1 |
|
evl1fpws.q |
|- O = ( eval1 ` R ) |
| 2 |
|
evl1fpws.w |
|- W = ( Poly1 ` R ) |
| 3 |
|
evl1fpws.b |
|- B = ( Base ` R ) |
| 4 |
|
evl1fpws.u |
|- U = ( Base ` W ) |
| 5 |
|
evl1fpws.s |
|- ( ph -> R e. CRing ) |
| 6 |
|
evl1fpws.y |
|- ( ph -> M e. U ) |
| 7 |
|
evl1fpws.1 |
|- .x. = ( .r ` R ) |
| 8 |
|
evl1fpws.2 |
|- .^ = ( .g ` ( mulGrp ` R ) ) |
| 9 |
|
evl1fpws.a |
|- A = ( coe1 ` M ) |
| 10 |
1 3
|
evl1fval1 |
|- O = ( R evalSub1 B ) |
| 11 |
10
|
fveq1i |
|- ( O ` M ) = ( ( R evalSub1 B ) ` M ) |
| 12 |
|
eqid |
|- ( R evalSub1 B ) = ( R evalSub1 B ) |
| 13 |
|
eqid |
|- ( Poly1 ` ( R |`s B ) ) = ( Poly1 ` ( R |`s B ) ) |
| 14 |
|
eqid |
|- ( R |`s B ) = ( R |`s B ) |
| 15 |
|
eqid |
|- ( Base ` ( Poly1 ` ( R |`s B ) ) ) = ( Base ` ( Poly1 ` ( R |`s B ) ) ) |
| 16 |
5
|
crngringd |
|- ( ph -> R e. Ring ) |
| 17 |
3
|
subrgid |
|- ( R e. Ring -> B e. ( SubRing ` R ) ) |
| 18 |
16 17
|
syl |
|- ( ph -> B e. ( SubRing ` R ) ) |
| 19 |
3
|
ressid |
|- ( R e. CRing -> ( R |`s B ) = R ) |
| 20 |
5 19
|
syl |
|- ( ph -> ( R |`s B ) = R ) |
| 21 |
20
|
fveq2d |
|- ( ph -> ( Poly1 ` ( R |`s B ) ) = ( Poly1 ` R ) ) |
| 22 |
21 2
|
eqtr4di |
|- ( ph -> ( Poly1 ` ( R |`s B ) ) = W ) |
| 23 |
22
|
fveq2d |
|- ( ph -> ( Base ` ( Poly1 ` ( R |`s B ) ) ) = ( Base ` W ) ) |
| 24 |
23 4
|
eqtr4di |
|- ( ph -> ( Base ` ( Poly1 ` ( R |`s B ) ) ) = U ) |
| 25 |
6 24
|
eleqtrrd |
|- ( ph -> M e. ( Base ` ( Poly1 ` ( R |`s B ) ) ) ) |
| 26 |
12 3 13 14 15 5 18 25 7 8 9
|
evls1fpws |
|- ( ph -> ( ( R evalSub1 B ) ` M ) = ( x e. B |-> ( R gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ x ) ) ) ) ) ) |
| 27 |
11 26
|
eqtrid |
|- ( ph -> ( O ` M ) = ( x e. B |-> ( R gsum ( k e. NN0 |-> ( ( A ` k ) .x. ( k .^ x ) ) ) ) ) ) |