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Theorem dvply2g 15448

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 15417	. . . 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 15419	. . . . . . . . . 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 5689	. . . . . . . 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 6825	. . . . . . . . . . . . 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 14532	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
14	13	subrgss 14194	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ⊆ ℂ)
15		0cn 8146	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
16		snssi 3812	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → {0} ⊆ ℂ)
18	14, 17	unssd 3380	. . . . . . . . . . . 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 5486	. . . . . . . . . 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 9474	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℤ)
25	24	uzidd 9745	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ (ℤ_≥‘𝑑))
26		peano2uz 9786	. . . . . . . . . . . 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 15439	. . . . . . . 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 4174	. . . . . . 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 2262	. . . . . 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 9432	. . . . . . . . . . 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 8170	. . . . . . . . . . 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 8466	. . . . . . . . . 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 2235	. . . . . . . . 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 6023	. . . . . . . 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 11885	. . . . . . 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 4175	. . . . . 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 6014	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
42		fvoveq1 6030	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑝‘(𝑐 + 1)) = (𝑝‘(𝑏 + 1)))
43	41, 42	oveq12d 6025	. . . . . . 7 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) = ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
44	43	cbvmptv 4180	. . . . . 6 ⊢ (𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1)))) = (𝑏 ∈ ℕ₀ ↦ ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
45		peano2nn0 9417	. . . . . . 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 15447	. . . . 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 10318	. . . . . . 7 ⊢ (𝑏 ∈ (0...𝑑) → 𝑏 ∈ ℕ₀)
50		peano2nn0 9417	. . . . . . . . . . . . 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 9432	. . . . . . . . . . 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 5773	. . . . . . . . . . 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 8175	. . . . . . . . . . 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 6014	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑐 + 1) → (𝑢 · 𝑣) = ((𝑐 + 1) · 𝑣))
57		oveq2 6015	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑝‘(𝑐 + 1)) → ((𝑐 + 1) · 𝑣) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
58		eqid 2229	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
59	56, 57, 58	ovmpog 6145	. . . . . . . . . . 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 1271	. . . . . . . . . 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 14557	. . . . . . . . . . . . 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 9575	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 14199	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ∈ (SubGrp‘ℂ_fld))
68		cnfld0 14543	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 = (0_g‘ℂ_fld)
69	68	subg0cl 13727	. . . . . . . . . . . . . . . . . 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 3813	. . . . . . . . . . . . . . 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 3377	. . . . . . . . . . . . . . 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 5462	. . . . . . . . . . . . 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 5773	. . . . . . . . . . 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 14535	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (._r‘ℂ_fld)
79	78	subrgmcl 14205	. . . . . . . . . . 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 1271	. . . . . . . . . 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 2307	. . . . . . . . 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 5792	. . . . . . . 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 5772	. . . . . . 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 15423	. . . . 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 2306	. . . 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 2656	. 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 1395 ∈ wcel 2200 ∃wrex 2509 ∪ cun 3195 ⊆ wss 3197 {csn 3666 ↦ cmpt 4145 “ cima 4722 ⟶wf 5314 ‘cfv 5318 (class class class)co 6007 ∈ cmpo 6009 ↑_𝑚 cmap 6803 ℂcc 8005 0cc0 8007 1c1 8008 + caddc 8010 · cmul 8012 − cmin 8325 ℕ₀cn0 9377 ℤcz 9454 ℤ_≥cuz 9730 ...cfz 10212 ↑cexp 10768 Σcsu 11872 SubGrpcsubg 13712 SubRingcsubrg 14189 ℂ_fldccnfld 14528 D cdv 15337 Polycply 15410

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127 ax-addf 8129 ax-mulf 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-of 6224 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-frec 6543 df-1o 6568 df-oadd 6572 df-er 6688 df-map 6805 df-pm 6806 df-en 6896 df-dom 6897 df-fin 6898 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-7 9182 df-8 9183 df-9 9184 df-n0 9378 df-z 9455 df-dec 9587 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-fz 10213 df-fzo 10347 df-seqfrec 10678 df-exp 10769 df-ihash 11006 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-clim 11798 df-sumdc 11873 df-struct 13042 df-ndx 13043 df-slot 13044 df-base 13046 df-sets 13047 df-iress 13048 df-plusg 13131 df-mulr 13132 df-starv 13133 df-tset 13137 df-ple 13138 df-ds 13140 df-unif 13141 df-rest 13282 df-topn 13283 df-0g 13299 df-topgen 13301 df-mgm 13397 df-sgrp 13443 df-mnd 13458 df-grp 13544 df-minusg 13545 df-mulg 13665 df-subg 13715 df-cmn 13831 df-mgp 13892 df-ur 13931 df-ring 13969 df-cring 13970 df-subrg 14191 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-fg 14521 df-metu 14522 df-cnfld 14529 df-top 14680 df-topon 14693 df-topsp 14713 df-bases 14725 df-ntr 14778 df-cn 14870 df-cnp 14871 df-tx 14935 df-xms 15021 df-ms 15022 df-cncf 15253 df-limced 15338 df-dvap 15339 df-ply 15412

This theorem is referenced by: dvply2 15449

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