Step |
Hyp |
Ref |
Expression |
1 |
|
ply1val.1 |
|- P = ( Poly1 ` R ) |
2 |
|
eqid |
|- ( PwSer1 ` R ) = ( PwSer1 ` R ) |
3 |
2
|
psr1crng |
|- ( R e. CRing -> ( PwSer1 ` R ) e. CRing ) |
4 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
5 |
1 2 4
|
ply1bas |
|- ( Base ` P ) = ( Base ` ( 1o mPoly R ) ) |
6 |
|
crngring |
|- ( R e. CRing -> R e. Ring ) |
7 |
1 2 4
|
ply1subrg |
|- ( R e. Ring -> ( Base ` P ) e. ( SubRing ` ( PwSer1 ` R ) ) ) |
8 |
6 7
|
syl |
|- ( R e. CRing -> ( Base ` P ) e. ( SubRing ` ( PwSer1 ` R ) ) ) |
9 |
5 8
|
eqeltrrid |
|- ( R e. CRing -> ( Base ` ( 1o mPoly R ) ) e. ( SubRing ` ( PwSer1 ` R ) ) ) |
10 |
1 2
|
ply1val |
|- P = ( ( PwSer1 ` R ) |`s ( Base ` ( 1o mPoly R ) ) ) |
11 |
10
|
subrgcrng |
|- ( ( ( PwSer1 ` R ) e. CRing /\ ( Base ` ( 1o mPoly R ) ) e. ( SubRing ` ( PwSer1 ` R ) ) ) -> P e. CRing ) |
12 |
3 9 11
|
syl2anc |
|- ( R e. CRing -> P e. CRing ) |