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Metamath Proof Explorer

Theorem mpfsubrg

Description: Polynomial functions are a subring. (Contributed by Mario Carneiro, 19-Mar-2015) (Revised by Mario Carneiro, 6-May-2015) (Revised by AV, 19-Sep-2021)

Ref Expression
Hypothesis mpfsubrg.q ⊢ 𝑄 = ran ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 )
Assertion mpfsubrg ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → 𝑄 ∈ ( SubRing ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )

Proof

Step Hyp Ref Expression
1 mpfsubrg.q ⊢ 𝑄 = ran ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 )
2 eqid ⊢ ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) = ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 )
3 eqid ⊢ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) = ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) )
4 eqid ⊢ ( 𝑆 ↾s 𝑅 ) = ( 𝑆 ↾s 𝑅 )
5 eqid ⊢ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) = ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) )
6 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
7 2 3 4 5 6 evlsrhm ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) ∈ ( ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) RingHom ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )
8 eqid ⊢ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) = ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) )
9 eqid ⊢ ( Base ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) = ( Base ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) )
10 8 9 rhmf ⊢ ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) ∈ ( ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) RingHom ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) → ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) : ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ⟶ ( Base ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )
11 ffn ⊢ ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) : ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ⟶ ( Base ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) → ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) Fn ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) )
12 fnima ⊢ ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) Fn ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) → ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) “ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) = ran ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) )
13 7 10 11 12 4syl ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) “ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) = ran ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) )
14 1 13 eqtr4id ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → 𝑄 = ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) “ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) )
15 4 subrgring ⊢ ( 𝑅 ∈ ( SubRing ‘ 𝑆 ) → ( 𝑆 ↾s 𝑅 ) ∈ Ring )
16 3 mplring ⊢ ( ( 𝐼 ∈ 𝑉 ∧ ( 𝑆 ↾s 𝑅 ) ∈ Ring ) → ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ∈ Ring )
17 15 16 sylan2 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ∈ Ring )
18 17 3adant2 ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ∈ Ring )
19 8 subrgid ⊢ ( ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ∈ Ring → ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ∈ ( SubRing ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) )
20 18 19 syl ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ∈ ( SubRing ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) )
21 rhmima ⊢ ( ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) ∈ ( ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) RingHom ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) ∧ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ∈ ( SubRing ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) → ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) “ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) ∈ ( SubRing ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )
22 7 20 21 syl2anc ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → ( ( ( 𝐼 evalSub 𝑆 ) ‘ 𝑅 ) “ ( Base ‘ ( 𝐼 mPoly ( 𝑆 ↾s 𝑅 ) ) ) ) ∈ ( SubRing ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )
23 14 22 eqeltrd ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → 𝑄 ∈ ( SubRing ‘ ( 𝑆 ↑s ( ( Base ‘ 𝑆 ) ↑m 𝐼 ) ) ) )