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Metamath Proof Explorer

Theorem rhmply1

Description: Provide a ring homomorphism between two univariate polynomial algebras over their respective base rings given a ring homomorphism between the two base rings. (Contributed by SN, 20-May-2025)

Ref Expression
Hypotheses rhmply1.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
rhmply1.q ⊢ 𝑄 = ( Poly1 ‘ 𝑆 )
rhmply1.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
rhmply1.f ⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) )
rhmply1.h ⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) )
Assertion rhmply1 ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑄 ) )

Proof

Step Hyp Ref Expression
1 rhmply1.p ⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2 rhmply1.q ⊢ 𝑄 = ( Poly1 ‘ 𝑆 )
3 rhmply1.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 rhmply1.f ⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) )
5 rhmply1.h ⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) )
6 eqid ⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 )
7 eqid ⊢ ( 1o mPoly 𝑆 ) = ( 1o mPoly 𝑆 )
8 1 3 ply1bas ⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) )
9 1oex ⊢ 1o ∈ V
10 9 a1i ⊢ ( 𝜑 → 1o ∈ V )
11 6 7 8 4 10 5 rhmmpl ⊢ ( 𝜑 → 𝐹 ∈ ( ( 1o mPoly 𝑅 ) RingHom ( 1o mPoly 𝑆 ) ) )
12 3 a1i ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) )
13 eqid ⊢ ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 )
14 13 a1i ⊢ ( 𝜑 → ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) )
15 8 a1i ⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) ) )
16 2 13 ply1bas ⊢ ( Base ‘ 𝑄 ) = ( Base ‘ ( 1o mPoly 𝑆 ) )
17 16 a1i ⊢ ( 𝜑 → ( Base ‘ 𝑄 ) = ( Base ‘ ( 1o mPoly 𝑆 ) ) )
18 eqid ⊢ ( +g ‘ 𝑃 ) = ( +g ‘ 𝑃 )
19 1 6 18 ply1plusg ⊢ ( +g ‘ 𝑃 ) = ( +g ‘ ( 1o mPoly 𝑅 ) )
20 19 oveqi ⊢ ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 1o mPoly 𝑅 ) ) 𝑦 )
21 20 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 1o mPoly 𝑅 ) ) 𝑦 ) )
22 eqid ⊢ ( +g ‘ 𝑄 ) = ( +g ‘ 𝑄 )
23 2 7 22 ply1plusg ⊢ ( +g ‘ 𝑄 ) = ( +g ‘ ( 1o mPoly 𝑆 ) )
24 23 oveqi ⊢ ( 𝑥 ( +g ‘ 𝑄 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 1o mPoly 𝑆 ) ) 𝑦 )
25 24 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( 𝑥 ( +g ‘ 𝑄 ) 𝑦 ) = ( 𝑥 ( +g ‘ ( 1o mPoly 𝑆 ) ) 𝑦 ) )
26 eqid ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ 𝑃 )
27 1 6 26 ply1mulr ⊢ ( .r ‘ 𝑃 ) = ( .r ‘ ( 1o mPoly 𝑅 ) )
28 27 oveqi ⊢ ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 1o mPoly 𝑅 ) ) 𝑦 )
29 28 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( .r ‘ 𝑃 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 1o mPoly 𝑅 ) ) 𝑦 ) )
30 eqid ⊢ ( .r ‘ 𝑄 ) = ( .r ‘ 𝑄 )
31 2 7 30 ply1mulr ⊢ ( .r ‘ 𝑄 ) = ( .r ‘ ( 1o mPoly 𝑆 ) )
32 31 oveqi ⊢ ( 𝑥 ( .r ‘ 𝑄 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 1o mPoly 𝑆 ) ) 𝑦 )
33 32 a1i ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ 𝑄 ) ∧ 𝑦 ∈ ( Base ‘ 𝑄 ) ) ) → ( 𝑥 ( .r ‘ 𝑄 ) 𝑦 ) = ( 𝑥 ( .r ‘ ( 1o mPoly 𝑆 ) ) 𝑦 ) )
34 12 14 15 17 21 25 29 33 rhmpropd ⊢ ( 𝜑 → ( 𝑃 RingHom 𝑄 ) = ( ( 1o mPoly 𝑅 ) RingHom ( 1o mPoly 𝑆 ) ) )
35 11 34 eleqtrrd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑄 ) )