Description: The ring of univariate polynomials over an integral domain is itself an integral domain. (Contributed by Stefan O'Rear, 29-Mar-2015)
Ref | Expression | ||
---|---|---|---|
Hypothesis | ply1domn.p | |- P = ( Poly1 ` R ) |
|
Assertion | ply1idom | |- ( R e. IDomn -> P e. IDomn ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ply1domn.p | |- P = ( Poly1 ` R ) |
|
2 | 1 | ply1crng | |- ( R e. CRing -> P e. CRing ) |
3 | 1 | ply1domn | |- ( R e. Domn -> P e. Domn ) |
4 | 2 3 | anim12i | |- ( ( R e. CRing /\ R e. Domn ) -> ( P e. CRing /\ P e. Domn ) ) |
5 | isidom | |- ( R e. IDomn <-> ( R e. CRing /\ R e. Domn ) ) |
|
6 | isidom | |- ( P e. IDomn <-> ( P e. CRing /\ P e. Domn ) ) |
|
7 | 4 5 6 | 3imtr4i | |- ( R e. IDomn -> P e. IDomn ) |