Step |
Hyp |
Ref |
Expression |
1 |
|
ply1domn.p |
|- P = ( Poly1 ` R ) |
2 |
|
nzrring |
|- ( R e. NzRing -> R e. Ring ) |
3 |
1
|
ply1ring |
|- ( R e. Ring -> P e. Ring ) |
4 |
2 3
|
syl |
|- ( R e. NzRing -> P e. Ring ) |
5 |
|
eqid |
|- ( algSc ` P ) = ( algSc ` P ) |
6 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
7 |
|
eqid |
|- ( Base ` P ) = ( Base ` P ) |
8 |
1 5 6 7
|
ply1sclf |
|- ( R e. Ring -> ( algSc ` P ) : ( Base ` R ) --> ( Base ` P ) ) |
9 |
2 8
|
syl |
|- ( R e. NzRing -> ( algSc ` P ) : ( Base ` R ) --> ( Base ` P ) ) |
10 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
11 |
6 10
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
12 |
2 11
|
syl |
|- ( R e. NzRing -> ( 1r ` R ) e. ( Base ` R ) ) |
13 |
9 12
|
ffvelrnd |
|- ( R e. NzRing -> ( ( algSc ` P ) ` ( 1r ` R ) ) e. ( Base ` P ) ) |
14 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
15 |
10 14
|
nzrnz |
|- ( R e. NzRing -> ( 1r ` R ) =/= ( 0g ` R ) ) |
16 |
|
eqid |
|- ( 0g ` P ) = ( 0g ` P ) |
17 |
1 5 14 16 6
|
ply1scln0 |
|- ( ( R e. Ring /\ ( 1r ` R ) e. ( Base ` R ) /\ ( 1r ` R ) =/= ( 0g ` R ) ) -> ( ( algSc ` P ) ` ( 1r ` R ) ) =/= ( 0g ` P ) ) |
18 |
2 12 15 17
|
syl3anc |
|- ( R e. NzRing -> ( ( algSc ` P ) ` ( 1r ` R ) ) =/= ( 0g ` P ) ) |
19 |
|
eldifsn |
|- ( ( ( algSc ` P ) ` ( 1r ` R ) ) e. ( ( Base ` P ) \ { ( 0g ` P ) } ) <-> ( ( ( algSc ` P ) ` ( 1r ` R ) ) e. ( Base ` P ) /\ ( ( algSc ` P ) ` ( 1r ` R ) ) =/= ( 0g ` P ) ) ) |
20 |
13 18 19
|
sylanbrc |
|- ( R e. NzRing -> ( ( algSc ` P ) ` ( 1r ` R ) ) e. ( ( Base ` P ) \ { ( 0g ` P ) } ) ) |
21 |
16 7
|
ringelnzr |
|- ( ( P e. Ring /\ ( ( algSc ` P ) ` ( 1r ` R ) ) e. ( ( Base ` P ) \ { ( 0g ` P ) } ) ) -> P e. NzRing ) |
22 |
4 20 21
|
syl2anc |
|- ( R e. NzRing -> P e. NzRing ) |