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Theorem dvply2g 15282

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 15251	. . . 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 15253	. . . . . . . . . 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 5639	. . . . . . . 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 6764	. . . . . . . . . . . . 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 14366	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
14	13	subrgss 14028	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ⊆ ℂ)
15		0cn 8071	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
16		snssi 3779	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → {0} ⊆ ℂ)
18	14, 17	unssd 3350	. . . . . . . . . . . 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 5444	. . . . . . . . . 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 9399	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℤ)
25	24	uzidd 9670	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ (ℤ_≥‘𝑑))
26		peano2uz 9711	. . . . . . . . . . . 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 15273	. . . . . . . 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 4138	. . . . . . 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 2239	. . . . . 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 9357	. . . . . . . . . . 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 8095	. . . . . . . . . . 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 8391	. . . . . . . . . 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 2212	. . . . . . . . 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 5967	. . . . . . . 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 11721	. . . . . . 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 4139	. . . . . 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 5958	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
42		fvoveq1 5974	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑝‘(𝑐 + 1)) = (𝑝‘(𝑏 + 1)))
43	41, 42	oveq12d 5969	. . . . . . 7 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) = ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
44	43	cbvmptv 4144	. . . . . 6 ⊢ (𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1)))) = (𝑏 ∈ ℕ₀ ↦ ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
45		peano2nn0 9342	. . . . . . 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 15281	. . . . 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 10243	. . . . . . 7 ⊢ (𝑏 ∈ (0...𝑑) → 𝑏 ∈ ℕ₀)
50		peano2nn0 9342	. . . . . . . . . . . . 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 9357	. . . . . . . . . . 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 5723	. . . . . . . . . . 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 8100	. . . . . . . . . . 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 5958	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑐 + 1) → (𝑢 · 𝑣) = ((𝑐 + 1) · 𝑣))
57		oveq2 5959	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑝‘(𝑐 + 1)) → ((𝑐 + 1) · 𝑣) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
58		eqid 2206	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
59	56, 57, 58	ovmpog 6087	. . . . . . . . . . 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 1250	. . . . . . . . . 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 14391	. . . . . . . . . . . . 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 9500	. . . . . . . . . . . 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 3195	. . . . . . . . . . 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 14033	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ∈ (SubGrp‘ℂ_fld))
68		cnfld0 14377	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 = (0_g‘ℂ_fld)
69	68	subg0cl 13562	. . . . . . . . . . . . . . . . . 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 3780	. . . . . . . . . . . . . . 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 3347	. . . . . . . . . . . . . . 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 5420	. . . . . . . . . . . . 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 5723	. . . . . . . . . . 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 14369	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (._r‘ℂ_fld)
79	78	subrgmcl 14039	. . . . . . . . . . 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 1250	. . . . . . . . . 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 2284	. . . . . . . . 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 5742	. . . . . . . 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 5722	. . . . . . 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 15257	. . . . 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 2283	. . . 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 2632	. 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 1373 ∈ wcel 2177 ∃wrex 2486 ∪ cun 3165 ⊆ wss 3167 {csn 3634 ↦ cmpt 4109 “ cima 4682 ⟶wf 5272 ‘cfv 5276 (class class class)co 5951 ∈ cmpo 5953 ↑_𝑚 cmap 6742 ℂcc 7930 0cc0 7932 1c1 7933 + caddc 7935 · cmul 7937 − cmin 8250 ℕ₀cn0 9302 ℤcz 9379 ℤ_≥cuz 9655 ...cfz 10137 ↑cexp 10690 Σcsu 11708 SubGrpcsubg 13547 SubRingcsubrg 14023 ℂ_fldccnfld 14362 D cdv 15171 Polycply 15244

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052 ax-addf 8054 ax-mulf 8055

This theorem depends on definitions: df-bi 117 df-stab 833 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-tp 3642 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-isom 5285 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-of 6165 df-1st 6233 df-2nd 6234 df-recs 6398 df-irdg 6463 df-frec 6484 df-1o 6509 df-oadd 6513 df-er 6627 df-map 6744 df-pm 6745 df-en 6835 df-dom 6836 df-fin 6837 df-sup 7093 df-inf 7094 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-5 9105 df-6 9106 df-7 9107 df-8 9108 df-9 9109 df-n0 9303 df-z 9380 df-dec 9512 df-uz 9656 df-q 9748 df-rp 9783 df-xneg 9901 df-xadd 9902 df-fz 10138 df-fzo 10272 df-seqfrec 10600 df-exp 10691 df-ihash 10928 df-cj 11197 df-re 11198 df-im 11199 df-rsqrt 11353 df-abs 11354 df-clim 11634 df-sumdc 11709 df-struct 12878 df-ndx 12879 df-slot 12880 df-base 12882 df-sets 12883 df-iress 12884 df-plusg 12966 df-mulr 12967 df-starv 12968 df-tset 12972 df-ple 12973 df-ds 12975 df-unif 12976 df-rest 13117 df-topn 13118 df-0g 13134 df-topgen 13136 df-mgm 13232 df-sgrp 13278 df-mnd 13293 df-grp 13379 df-minusg 13380 df-mulg 13500 df-subg 13550 df-cmn 13666 df-mgp 13727 df-ur 13766 df-ring 13804 df-cring 13805 df-subrg 14025 df-psmet 14349 df-xmet 14350 df-met 14351 df-bl 14352 df-mopn 14353 df-fg 14355 df-metu 14356 df-cnfld 14363 df-top 14514 df-topon 14527 df-topsp 14547 df-bases 14559 df-ntr 14612 df-cn 14704 df-cnp 14705 df-tx 14769 df-xms 14855 df-ms 14856 df-cncf 15087 df-limced 15172 df-dvap 15173 df-ply 15246

This theorem is referenced by: dvply2 15283

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