# Metamath Proof Explorer

## Theorem pf1subrg

Description: Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses pf1const.b 𝐵 = ( Base ‘ 𝑅 )
pf1const.q 𝑄 = ran ( eval1𝑅 )
Assertion pf1subrg ( 𝑅 ∈ CRing → 𝑄 ∈ ( SubRing ‘ ( 𝑅s 𝐵 ) ) )

### Proof

Step Hyp Ref Expression
1 pf1const.b 𝐵 = ( Base ‘ 𝑅 )
2 pf1const.q 𝑄 = ran ( eval1𝑅 )
3 eqid ( eval1𝑅 ) = ( eval1𝑅 )
4 eqid ( Poly1𝑅 ) = ( Poly1𝑅 )
5 eqid ( 𝑅s 𝐵 ) = ( 𝑅s 𝐵 )
6 3 4 5 1 evl1rhm ( 𝑅 ∈ CRing → ( eval1𝑅 ) ∈ ( ( Poly1𝑅 ) RingHom ( 𝑅s 𝐵 ) ) )
7 eqid ( Base ‘ ( Poly1𝑅 ) ) = ( Base ‘ ( Poly1𝑅 ) )
8 eqid ( Base ‘ ( 𝑅s 𝐵 ) ) = ( Base ‘ ( 𝑅s 𝐵 ) )
9 7 8 rhmf ( ( eval1𝑅 ) ∈ ( ( Poly1𝑅 ) RingHom ( 𝑅s 𝐵 ) ) → ( eval1𝑅 ) : ( Base ‘ ( Poly1𝑅 ) ) ⟶ ( Base ‘ ( 𝑅s 𝐵 ) ) )
10 ffn ( ( eval1𝑅 ) : ( Base ‘ ( Poly1𝑅 ) ) ⟶ ( Base ‘ ( 𝑅s 𝐵 ) ) → ( eval1𝑅 ) Fn ( Base ‘ ( Poly1𝑅 ) ) )
11 fnima ( ( eval1𝑅 ) Fn ( Base ‘ ( Poly1𝑅 ) ) → ( ( eval1𝑅 ) “ ( Base ‘ ( Poly1𝑅 ) ) ) = ran ( eval1𝑅 ) )
12 6 9 10 11 4syl ( 𝑅 ∈ CRing → ( ( eval1𝑅 ) “ ( Base ‘ ( Poly1𝑅 ) ) ) = ran ( eval1𝑅 ) )
13 12 2 eqtr4di ( 𝑅 ∈ CRing → ( ( eval1𝑅 ) “ ( Base ‘ ( Poly1𝑅 ) ) ) = 𝑄 )
14 4 ply1assa ( 𝑅 ∈ CRing → ( Poly1𝑅 ) ∈ AssAlg )
15 assaring ( ( Poly1𝑅 ) ∈ AssAlg → ( Poly1𝑅 ) ∈ Ring )
16 7 subrgid ( ( Poly1𝑅 ) ∈ Ring → ( Base ‘ ( Poly1𝑅 ) ) ∈ ( SubRing ‘ ( Poly1𝑅 ) ) )
17 14 15 16 3syl ( 𝑅 ∈ CRing → ( Base ‘ ( Poly1𝑅 ) ) ∈ ( SubRing ‘ ( Poly1𝑅 ) ) )
18 rhmima ( ( ( eval1𝑅 ) ∈ ( ( Poly1𝑅 ) RingHom ( 𝑅s 𝐵 ) ) ∧ ( Base ‘ ( Poly1𝑅 ) ) ∈ ( SubRing ‘ ( Poly1𝑅 ) ) ) → ( ( eval1𝑅 ) “ ( Base ‘ ( Poly1𝑅 ) ) ) ∈ ( SubRing ‘ ( 𝑅s 𝐵 ) ) )
19 6 17 18 syl2anc ( 𝑅 ∈ CRing → ( ( eval1𝑅 ) “ ( Base ‘ ( Poly1𝑅 ) ) ) ∈ ( SubRing ‘ ( 𝑅s 𝐵 ) ) )
20 13 19 eqeltrrd ( 𝑅 ∈ CRing → 𝑄 ∈ ( SubRing ‘ ( 𝑅s 𝐵 ) ) )