This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem evlvvvallem

Description: Lemma for theorems using evlvvval . Version of evlsvvvallem2 using df-evl . (Contributed by SN, 11-Mar-2025)

Ref Expression
Hypotheses evlvvvallem.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
evlvvvallem.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
evlvvvallem.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
evlvvvallem.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
evlvvvallem.m ⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
evlvvvallem.w ⊢ ↑ = ( .g ‘ 𝑀 )
evlvvvallem.x ⊢ · = ( .r ‘ 𝑅 )
evlvvvallem.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
evlvvvallem.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
evlvvvallem.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
evlvvvallem.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
Assertion evlvvvallem ( 𝜑 → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ 𝑏 ) · ( 𝑀 Σg ( 𝑣 ∈ 𝐼 ↦ ( ( 𝑏 ‘ 𝑣 ) ↑ ( 𝐴 ‘ 𝑣 ) ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 evlvvvallem.d ⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
2 evlvvvallem.p ⊢ 𝑃 = ( 𝐼 mPoly 𝑅 )
3 evlvvvallem.b ⊢ 𝐵 = ( Base ‘ 𝑃 )
4 evlvvvallem.k ⊢ 𝐾 = ( Base ‘ 𝑅 )
5 evlvvvallem.m ⊢ 𝑀 = ( mulGrp ‘ 𝑅 )
6 evlvvvallem.w ⊢ ↑ = ( .g ‘ 𝑀 )
7 evlvvvallem.x ⊢ · = ( .r ‘ 𝑅 )
8 evlvvvallem.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
9 evlvvvallem.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
10 evlvvvallem.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
11 evlvvvallem.a ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐾 ↑m 𝐼 ) )
12 eqid ⊢ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) = ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) )
13 eqid ⊢ ( 𝑅 ↾s 𝐾 ) = ( 𝑅 ↾s 𝐾 )
14 eqid ⊢ ( Base ‘ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) ) = ( Base ‘ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) )
15 9 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
16 4 subrgid ⊢ ( 𝑅 ∈ Ring → 𝐾 ∈ ( SubRing ‘ 𝑅 ) )
17 15 16 syl ⊢ ( 𝜑 → 𝐾 ∈ ( SubRing ‘ 𝑅 ) )
18 4 ressid ⊢ ( 𝑅 ∈ CRing → ( 𝑅 ↾s 𝐾 ) = 𝑅 )
19 9 18 syl ⊢ ( 𝜑 → ( 𝑅 ↾s 𝐾 ) = 𝑅 )
20 19 oveq2d ⊢ ( 𝜑 → ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) = ( 𝐼 mPoly 𝑅 ) )
21 20 2 eqtr4di ⊢ ( 𝜑 → ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) = 𝑃 )
22 21 fveq2d ⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) ) = ( Base ‘ 𝑃 ) )
23 22 3 eqtr4di ⊢ ( 𝜑 → ( Base ‘ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) ) = 𝐵 )
24 10 23 eleqtrrd ⊢ ( 𝜑 → 𝐹 ∈ ( Base ‘ ( 𝐼 mPoly ( 𝑅 ↾s 𝐾 ) ) ) )
25 1 12 13 14 4 5 6 7 8 9 17 24 11 evlsvvvallem2 ⊢ ( 𝜑 → ( 𝑏 ∈ 𝐷 ↦ ( ( 𝐹 ‘ 𝑏 ) · ( 𝑀 Σg ( 𝑣 ∈ 𝐼 ↦ ( ( 𝑏 ‘ 𝑣 ) ↑ ( 𝐴 ‘ 𝑣 ) ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) )