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Metamath Proof Explorer

Theorem evlcl

Description: A polynomial over the ring R evaluates to an element in R . (Contributed by SN, 12-Mar-2025)

Ref Expression
Hypotheses evlcl.q ⊢ 𝑄 = ( 𝐼 eval 𝑅 )
evlcl.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
evlcl.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
evlcl.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
evlcl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
evlcl.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
evlcl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
evlcl.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
Assertion evlcl ( 𝜑 → ( ( 𝑄 ‘ 𝐹 ) ‘ 𝐴 ) ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 evlcl.q ⊢ 𝑄 = ( 𝐼 eval 𝑅 )
2 evlcl.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3 evlcl.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 evlcl.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
5 evlcl.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6 evlcl.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
7 evlcl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 evlcl.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
9 eqid ⊢ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) = ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) )
10 eqid ⊢ ( Base ‘ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) = ( Base ‘ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) )
11 ovexd ⊢ ( 𝜑 → ( 𝐾 ↑m 𝐼 ) ∈ V )
12 1 4 2 9 evlrhm ⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ CRing ) → 𝑄 ∈ ( 𝑃 RingHom ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
13 5 6 12 syl2anc ⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 RingHom ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
14 3 10 rhmf ⊢ ( 𝑄 ∈ ( 𝑃 RingHom ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) → 𝑄 : 𝐵 ⟶ ( Base ‘ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
15 13 14 syl ⊢ ( 𝜑 → 𝑄 : 𝐵 ⟶ ( Base ‘ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
16 15 7 ffvelcdmd ⊢ ( 𝜑 → ( 𝑄 ‘ 𝐹 ) ∈ ( Base ‘ ( 𝑅 ↑s ( 𝐾 ↑m 𝐼 ) ) ) )
17 9 4 10 6 11 16 pwselbas ⊢ ( 𝜑 → ( 𝑄 ‘ 𝐹 ) : ( 𝐾 ↑m 𝐼 ) ⟶ 𝐾 )
18 17 8 ffvelcdmd ⊢ ( 𝜑 → ( ( 𝑄 ‘ 𝐹 ) ‘ 𝐴 ) ∈ 𝐾 )