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Metamath Proof Explorer

Theorem evl1maprhm

Description: The function F mapping polynomials p to their evaluation at a given point X is a ring homomorphism. (Contributed by metakunt, 19-May-2025)

Ref Expression
Hypotheses evl1maprhm.q ⊢ 𝑂 = ( eval1 ‘ 𝑅 )
evl1maprhm.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
evl1maprhm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
evl1maprhm.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
evl1maprhm.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
evl1maprhm.y ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
evl1maprhm.f ⊢ 𝐹 = ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) )
Assertion evl1maprhm ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑅 ) )

Proof

Step Hyp Ref Expression
1 evl1maprhm.q ⊢ 𝑂 = ( eval1 ‘ 𝑅 )
2 evl1maprhm.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3 evl1maprhm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
4 evl1maprhm.u ⊢ 𝑈 = ( Base ‘ 𝑃 )
5 evl1maprhm.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
6 evl1maprhm.y ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7 evl1maprhm.f ⊢ 𝐹 = ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) )
8 7 a1i ⊢ ( 𝜑 → 𝐹 = ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) ) )
9 ssidd ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) )
10 5 elexd ⊢ ( 𝜑 → 𝑅 ∈ V )
11 5 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
12 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
13 12 subrgid ⊢ ( 𝑅 ∈ Ring → ( Base ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) )
14 11 13 syl ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ∈ ( SubRing ‘ 𝑅 ) )
15 14 elexd ⊢ ( 𝜑 → ( Base ‘ 𝑅 ) ∈ V )
16 eqid ⊢ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = ( 𝑅 ↾s ( Base ‘ 𝑅 ) )
17 16 12 ressid2 ⊢ ( ( ( Base ‘ 𝑅 ) ⊆ ( Base ‘ 𝑅 ) ∧ 𝑅 ∈ V ∧ ( Base ‘ 𝑅 ) ∈ V ) → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 )
18 9 10 15 17 syl3anc ⊢ ( 𝜑 → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 )
19 eqcom ⊢ ( ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 ↔ 𝑅 = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
20 19 imbi2i ⊢ ( ( 𝜑 → ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) = 𝑅 ) ↔ ( 𝜑 → 𝑅 = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) )
21 18 20 mpbi ⊢ ( 𝜑 → 𝑅 = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
22 21 fveq2d ⊢ ( 𝜑 → ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) )
23 2 22 eqtrid ⊢ ( 𝜑 → 𝑃 = ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) )
24 23 fveq2d ⊢ ( 𝜑 → ( Base ‘ 𝑃 ) = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) )
25 4 24 eqtrid ⊢ ( 𝜑 → 𝑈 = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) )
26 1 12 evl1fval1 ⊢ 𝑂 = ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) )
27 26 a1i ⊢ ( 𝜑 → 𝑂 = ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) )
28 27 fveq1d ⊢ ( 𝜑 → ( 𝑂 ‘ 𝑝 ) = ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) )
29 28 fveq1d ⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) = ( ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) ‘ 𝑋 ) )
30 25 29 mpteq12dv ⊢ ( 𝜑 → ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) ) = ( 𝑝 ∈ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) ↦ ( ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) ‘ 𝑋 ) ) )
31 eqid ⊢ ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) = ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) )
32 eqid ⊢ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) = ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
33 eqid ⊢ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) )
34 6 3 eleqtrdi ⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝑅 ) )
35 eqid ⊢ ( 𝑝 ∈ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) ↦ ( ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) ‘ 𝑋 ) ) = ( 𝑝 ∈ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) ↦ ( ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) ‘ 𝑋 ) )
36 31 32 12 33 5 14 34 35 evls1maprhm ⊢ ( 𝜑 → ( 𝑝 ∈ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) ) ↦ ( ( ( 𝑅 evalSub1 ( Base ‘ 𝑅 ) ) ‘ 𝑝 ) ‘ 𝑋 ) ) ∈ ( ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) RingHom 𝑅 ) )
37 30 36 eqeltrd ⊢ ( 𝜑 → ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) ) ∈ ( ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) RingHom 𝑅 ) )
38 2 a1i ⊢ ( 𝜑 → 𝑃 = ( Poly1 ‘ 𝑅 ) )
39 18 eqcomd ⊢ ( 𝜑 → 𝑅 = ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) )
40 39 fveq2d ⊢ ( 𝜑 → ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) )
41 38 40 eqtr2d ⊢ ( 𝜑 → ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) = 𝑃 )
42 41 oveq1d ⊢ ( 𝜑 → ( ( Poly1 ‘ ( 𝑅 ↾s ( Base ‘ 𝑅 ) ) ) RingHom 𝑅 ) = ( 𝑃 RingHom 𝑅 ) )
43 37 42 eleqtrd ⊢ ( 𝜑 → ( 𝑝 ∈ 𝑈 ↦ ( ( 𝑂 ‘ 𝑝 ) ‘ 𝑋 ) ) ∈ ( 𝑃 RingHom 𝑅 ) )
44 8 43 eqeltrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑅 ) )