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Theorem dvply2g 15086

Description: The derivative of a polynomial with coefficients in a subring is a polynomial with coefficients in the same ring. (Contributed by Mario Carneiro, 1-Jan-2017.) (Revised by GG, 30-Apr-2025.)

**Assertion**
Ref	Expression
dvply2g	⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Proof of Theorem dvply2g Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑝 𝑢 𝑣 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elply2 15055	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) ↔ (𝑆 ⊆ ℂ ∧ ∃𝑑 ∈ ℕ₀ ∃𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀)((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))))
2	1	simprbi 275	. . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∃𝑑 ∈ ℕ₀ ∃𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀)((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘)))))
3	2	adantl 277	. 2 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → ∃𝑑 ∈ ℕ₀ ∃𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀)((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘)))))
4		plyf 15057	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
5	4	adantl 277	. . . . . . . . 9 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹:ℂ⟶ℂ)
6	5	feqmptd 5617	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)))
7	6	ad2antrr 488	. . . . . . 7 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝐹 = (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)))
8		simplrl 535	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑑 ∈ ℕ₀)
9	8	adantr 276	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → 𝑑 ∈ ℕ₀)
10		elmapi 6738	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) → 𝑝:ℕ₀⟶(𝑆 ∪ {0}))
11	10	ad2antll 491	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) → 𝑝:ℕ₀⟶(𝑆 ∪ {0}))
12	11	adantr 276	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑝:ℕ₀⟶(𝑆 ∪ {0}))
13		cnfldbas 14192	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
14	13	subrgss 13854	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ⊆ ℂ)
15		0cn 8035	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
16		snssi 3767	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → {0} ⊆ ℂ)
18	14, 17	unssd 3340	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → (𝑆 ∪ {0}) ⊆ ℂ)
19	18	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑆 ∪ {0}) ⊆ ℂ)
20	12, 19	fssd 5423	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑝:ℕ₀⟶ℂ)
21	20	adantr 276	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → 𝑝:ℕ₀⟶ℂ)
22		simplrl 535	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → (𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0})
23		simplrr 536	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))
24		nn0z 9363	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℤ)
25	24	uzidd 9633	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ (ℤ_≥‘𝑑))
26		peano2uz 9674	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ (ℤ_≥‘𝑑) → (𝑑 + 1) ∈ (ℤ_≥‘𝑑))
27	25, 26	syl 14	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ (ℤ_≥‘𝑑))
28	8, 27	syl 14	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑑 + 1) ∈ (ℤ_≥‘𝑑))
29	28	adantr 276	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → (𝑑 + 1) ∈ (ℤ_≥‘𝑑))
30		simpr 110	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → 𝑎 ∈ ℂ)
31	9, 21, 22, 23, 29, 30	plycoeid3 15077	. . . . . . . 8 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑎 ∈ ℂ) → (𝐹‘𝑎) = Σ𝑏 ∈ (0...(𝑑 + 1))((𝑝‘𝑏) · (𝑎↑𝑏)))
32	31	mpteq2dva 4124	. . . . . . 7 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(𝑑 + 1))((𝑝‘𝑏) · (𝑎↑𝑏))))
33	7, 32	eqtrd 2229	. . . . . 6 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝐹 = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...(𝑑 + 1))((𝑝‘𝑏) · (𝑎↑𝑏))))
34	8	nn0cnd 9321	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑑 ∈ ℂ)
35		1cnd 8059	. . . . . . . . . . 11 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 1 ∈ ℂ)
36	34, 35	pncand 8355	. . . . . . . . . 10 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → ((𝑑 + 1) − 1) = 𝑑)
37	36	eqcomd 2202	. . . . . . . . 9 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑑 = ((𝑑 + 1) − 1))
38	37	oveq2d 5941	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (0...𝑑) = (0...((𝑑 + 1) − 1)))
39	38	sumeq1d 11548	. . . . . . 7 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → Σ𝑏 ∈ (0...𝑑)(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏)) = Σ𝑏 ∈ (0...((𝑑 + 1) − 1))(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏)))
40	39	mpteq2dv 4125	. . . . . 6 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...𝑑)(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...((𝑑 + 1) − 1))(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))))
41		oveq1 5932	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
42		fvoveq1 5948	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑝‘(𝑐 + 1)) = (𝑝‘(𝑏 + 1)))
43	41, 42	oveq12d 5943	. . . . . . 7 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) = ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
44	43	cbvmptv 4130	. . . . . 6 ⊢ (𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1)))) = (𝑏 ∈ ℕ₀ ↦ ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
45		peano2nn0 9306	. . . . . . 7 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
46	8, 45	syl 14	. . . . . 6 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑑 + 1) ∈ ℕ₀)
47	33, 40, 20, 44, 46	dvply1 15085	. . . . 5 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (ℂ D 𝐹) = (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...𝑑)(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))))
48	14	ad3antrrr 492	. . . . . 6 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → 𝑆 ⊆ ℂ)
49		elfznn0 10206	. . . . . . 7 ⊢ (𝑏 ∈ (0...𝑑) → 𝑏 ∈ ℕ₀)
50		peano2nn0 9306	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ ℕ₀ → (𝑐 + 1) ∈ ℕ₀)
51	50	adantl 277	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑐 + 1) ∈ ℕ₀)
52	51	nn0cnd 9321	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑐 + 1) ∈ ℂ)
53	20	adantr 276	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → 𝑝:ℕ₀⟶ℂ)
54	53, 51	ffvelcdmd 5701	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑝‘(𝑐 + 1)) ∈ ℂ)
55	52, 54	mulcld 8064	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) ∈ ℂ)
56		oveq1 5932	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑐 + 1) → (𝑢 · 𝑣) = ((𝑐 + 1) · 𝑣))
57		oveq2 5933	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑝‘(𝑐 + 1)) → ((𝑐 + 1) · 𝑣) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
58		eqid 2196	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
59	56, 57, 58	ovmpog 6061	. . . . . . . . . . 11 ⊢ (((𝑐 + 1) ∈ ℂ ∧ (𝑝‘(𝑐 + 1)) ∈ ℂ ∧ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) ∈ ℂ) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑝‘(𝑐 + 1))) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
60	52, 54, 55, 59	syl3anc 1249	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑝‘(𝑐 + 1))) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
61		simp-4l 541	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → 𝑆 ∈ (SubRing‘ℂ_fld))
62		zsssubrg 14217	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → ℤ ⊆ 𝑆)
63	62	ad4antr 494	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → ℤ ⊆ 𝑆)
64	51	nn0zd 9463	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑐 + 1) ∈ ℤ)
65	63, 64	sseldd 3185	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑐 + 1) ∈ 𝑆)
66	12	adantr 276	. . . . . . . . . . . . 13 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → 𝑝:ℕ₀⟶(𝑆 ∪ {0}))
67		subrgsubg 13859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ∈ (SubGrp‘ℂ_fld))
68		cnfld0 14203	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 = (0_g‘ℂ_fld)
69	68	subg0cl 13388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubGrp‘ℂ_fld) → 0 ∈ 𝑆)
70	67, 69	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 0 ∈ 𝑆)
71	70	ad4antr 494	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → 0 ∈ 𝑆)
72	71	snssd 3768	. . . . . . . . . . . . . . 15 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → {0} ⊆ 𝑆)
73		ssequn2 3337	. . . . . . . . . . . . . . 15 ⊢ ({0} ⊆ 𝑆 ↔ (𝑆 ∪ {0}) = 𝑆)
74	72, 73	sylib 122	. . . . . . . . . . . . . 14 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑆 ∪ {0}) = 𝑆)
75	74	feq3d 5399	. . . . . . . . . . . . 13 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑝:ℕ₀⟶(𝑆 ∪ {0}) ↔ 𝑝:ℕ₀⟶𝑆))
76	66, 75	mpbid 147	. . . . . . . . . . . 12 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → 𝑝:ℕ₀⟶𝑆)
77	76, 51	ffvelcdmd 5701	. . . . . . . . . . 11 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → (𝑝‘(𝑐 + 1)) ∈ 𝑆)
78		mpocnfldmul 14195	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (._r‘ℂ_fld)
79	78	subrgmcl 13865	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ (𝑐 + 1) ∈ 𝑆 ∧ (𝑝‘(𝑐 + 1)) ∈ 𝑆) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑝‘(𝑐 + 1))) ∈ 𝑆)
80	61, 65, 77, 79	syl3anc 1249	. . . . . . . . . 10 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → ((𝑐 + 1)(𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))(𝑝‘(𝑐 + 1))) ∈ 𝑆)
81	60, 80	eqeltrrd 2274	. . . . . . . . 9 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑐 ∈ ℕ₀) → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) ∈ 𝑆)
82	81	fmpttd 5720	. . . . . . . 8 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1)))):ℕ₀⟶𝑆)
83	82	ffvelcdmda 5700	. . . . . . 7 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑏 ∈ ℕ₀) → ((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) ∈ 𝑆)
84	49, 83	sylan2 286	. . . . . 6 ⊢ (((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) ∧ 𝑏 ∈ (0...𝑑)) → ((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) ∈ 𝑆)
85	48, 8, 84	elplyd 15061	. . . . 5 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (𝑎 ∈ ℂ ↦ Σ𝑏 ∈ (0...𝑑)(((𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))‘𝑏) · (𝑎↑𝑏))) ∈ (Poly‘𝑆))
86	47, 85	eqeltrd 2273	. . . 4 ⊢ ((((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) ∧ ((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘))))) → (ℂ D 𝐹) ∈ (Poly‘𝑆))
87	86	ex 115	. . 3 ⊢ (((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) ∧ (𝑑 ∈ ℕ₀ ∧ 𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀))) → (((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘)))) → (ℂ D 𝐹) ∈ (Poly‘𝑆)))
88	87	rexlimdvva 2622	. 2 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (∃𝑑 ∈ ℕ₀ ∃𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀)((𝑝 “ (ℤ_≥‘(𝑑 + 1))) = {0} ∧ 𝐹 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝑝‘𝑘) · (𝑧↑𝑘)))) → (ℂ D 𝐹) ∈ (Poly‘𝑆)))
89	3, 88	mpd 13	1 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → (ℂ D 𝐹) ∈ (Poly‘𝑆))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2167 ∃wrex 2476 ∪ cun 3155 ⊆ wss 3157 {csn 3623 ↦ cmpt 4095 “ cima 4667 ⟶wf 5255 ‘cfv 5259 (class class class)co 5925 ∈ cmpo 5927 ↑_𝑚 cmap 6716 ℂcc 7894 0cc0 7896 1c1 7897 + caddc 7899 · cmul 7901 − cmin 8214 ℕ₀cn0 9266 ℤcz 9343 ℤ_≥cuz 9618 ...cfz 10100 ↑cexp 10647 Σcsu 11535 SubGrpcsubg 13373 SubRingcsubrg 13849 ℂ_fldccnfld 14188 D cdv 14975 Polycply 15048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016 ax-addf 8018 ax-mulf 8019

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-tp 3631 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-5 9069 df-6 9070 df-7 9071 df-8 9072 df-9 9073 df-n0 9267 df-z 9344 df-dec 9475 df-uz 9619 df-q 9711 df-rp 9746 df-xneg 9864 df-xadd 9865 df-fz 10101 df-fzo 10235 df-seqfrec 10557 df-exp 10648 df-ihash 10885 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181 df-clim 11461 df-sumdc 11536 df-struct 12705 df-ndx 12706 df-slot 12707 df-base 12709 df-sets 12710 df-iress 12711 df-plusg 12793 df-mulr 12794 df-starv 12795 df-tset 12799 df-ple 12800 df-ds 12802 df-unif 12803 df-rest 12943 df-topn 12944 df-0g 12960 df-topgen 12962 df-mgm 13058 df-sgrp 13104 df-mnd 13119 df-grp 13205 df-minusg 13206 df-mulg 13326 df-subg 13376 df-cmn 13492 df-mgp 13553 df-ur 13592 df-ring 13630 df-cring 13631 df-subrg 13851 df-psmet 14175 df-xmet 14176 df-met 14177 df-bl 14178 df-mopn 14179 df-fg 14181 df-metu 14182 df-cnfld 14189 df-top 14318 df-topon 14331 df-topsp 14351 df-bases 14363 df-ntr 14416 df-cn 14508 df-cnp 14509 df-tx 14573 df-xms 14659 df-ms 14660 df-cncf 14891 df-limced 14976 df-dvap 14977 df-ply 15050

This theorem is referenced by: dvply2 15087

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