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Theorem dvply2g 16004

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 15973	. . . 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 15975	. . . . . . . . . 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 5759	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubRing‘ℂ_fld) ∧ 𝐹 ∈ (Poly‘𝑆)) → 𝐹 = (𝑎 ∈ ℂ ↦ (𝐹‘𝑎)))
7	6	ad2antrr 492	. . . . . . 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 541	. . . . . . . . . 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 6948	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ((𝑆 ∪ {0}) ↑_𝑚 ℕ₀) → 𝑝:ℕ₀⟶(𝑆 ∪ {0}))
11	10	ad2antll 495	. . . . . . . . . . . 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 15007	. . . . . . . . . . . . . 14 ⊢ ℂ = (Base‘ℂ_fld)
14	13	subrgss 14640	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ⊆ ℂ)
15		0cn 8325	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
16		snssi 3859	. . . . . . . . . . . . . 14 ⊢ (0 ∈ ℂ → {0} ⊆ ℂ)
17	15, 16	mp1i 10	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → {0} ⊆ ℂ)
18	14, 17	unssd 3405	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → (𝑆 ∪ {0}) ⊆ ℂ)
19	18	ad3antrrr 496	. . . . . . . . . . 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 5550	. . . . . . . . . 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 541	. . . . . . . . 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 542	. . . . . . . . 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 9676	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ ℤ)
25	24	uzidd 9954	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → 𝑑 ∈ (ℤ_≥‘𝑑))
26		peano2uz 10000	. . . . . . . . . . . 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 15995	. . . . . . . 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 4223	. . . . . . 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 2271	. . . . . 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 9634	. . . . . . . . . . 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 8349	. . . . . . . . . . 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 8646	. . . . . . . . . 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 2244	. . . . . . . . 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 6104	. . . . . . . 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 12158	. . . . . . 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 4224	. . . . . 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 6095	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑐 + 1) = (𝑏 + 1))
42		fvoveq1 6111	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → (𝑝‘(𝑐 + 1)) = (𝑝‘(𝑏 + 1)))
43	41, 42	oveq12d 6106	. . . . . . 7 ⊢ (𝑐 = 𝑏 → ((𝑐 + 1) · (𝑝‘(𝑐 + 1))) = ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
44	43	cbvmptv 4229	. . . . . 6 ⊢ (𝑐 ∈ ℕ₀ ↦ ((𝑐 + 1) · (𝑝‘(𝑐 + 1)))) = (𝑏 ∈ ℕ₀ ↦ ((𝑏 + 1) · (𝑝‘(𝑏 + 1))))
45		peano2nn0 9615	. . . . . . 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 16003	. . . . 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 496	. . . . . 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 10539	. . . . . . 7 ⊢ (𝑏 ∈ (0...𝑑) → 𝑏 ∈ ℕ₀)
50		peano2nn0 9615	. . . . . . . . . . . . 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 9634	. . . . . . . . . . 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 5847	. . . . . . . . . . 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 8353	. . . . . . . . . . 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 6095	. . . . . . . . . . . 12 ⊢ (𝑢 = (𝑐 + 1) → (𝑢 · 𝑣) = ((𝑐 + 1) · 𝑣))
57		oveq2 6096	. . . . . . . . . . . 12 ⊢ (𝑣 = (𝑝‘(𝑐 + 1)) → ((𝑐 + 1) · 𝑣) = ((𝑐 + 1) · (𝑝‘(𝑐 + 1))))
58		eqid 2238	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))
59	56, 57, 58	ovmpog 6226	. . . . . . . . . . 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 1278	. . . . . . . . . 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 547	. . . . . . . . . . 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 15032	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → ℤ ⊆ 𝑆)
63	62	ad4antr 498	. . . . . . . . . . . 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 9778	. . . . . . . . . . . 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 3249	. . . . . . . . . . 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 14645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 𝑆 ∈ (SubGrp‘ℂ_fld))
68		cnfld0 15018	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 = (0_g‘ℂ_fld)
69	68	subg0cl 14045	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (SubGrp‘ℂ_fld) → 0 ∈ 𝑆)
70	67, 69	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ (SubRing‘ℂ_fld) → 0 ∈ 𝑆)
71	70	ad4antr 498	. . . . . . . . . . . . . . . 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 3860	. . . . . . . . . . . . . . 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 3402	. . . . . . . . . . . . . . 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 5525	. . . . . . . . . . . . 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 5847	. . . . . . . . . . 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 15010	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) = (._r‘ℂ_fld)
79	78	subrgmcl 14651	. . . . . . . . . . 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 1278	. . . . . . . . . 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 2316	. . . . . . . . 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 5866	. . . . . . . 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 5846	. . . . . . 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 15979	. . . . 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 2315	. . . 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 2676	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 ∪ cun 3218 ⊆ wss 3220 {csn 3709 ↦ cmpt 4193 “ cima 4779 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ∈ cmpo 6090 ↑_𝑚 cmap 6926 ℂcc 8184 0cc0 8186 1c1 8187 + caddc 8189 · cmul 8191 − cmin 8505 ℕ₀cn0 9575 ℤcz 9656 ℤ_≥cuz 9938 ...cfz 10429 ↑cexp 10997 Σcsu 12145 SubGrpcsubg 14030 SubRingcsubrg 14635 ℂ_fldccnfld 15003 D cdv 15893 Polycply 15966

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-mulrcl 8285 ax-addcom 8286 ax-mulcom 8287 ax-addass 8288 ax-mulass 8289 ax-distr 8290 ax-i2m1 8291 ax-0lt1 8292 ax-1rid 8293 ax-0id 8294 ax-rnegex 8295 ax-precex 8296 ax-cnre 8297 ax-pre-ltirr 8298 ax-pre-ltwlin 8299 ax-pre-lttrn 8300 ax-pre-apti 8301 ax-pre-ltadd 8302 ax-pre-mulgt0 8303 ax-pre-mulext 8304 ax-arch 8305 ax-caucvg 8306 ax-addf 8308 ax-mulf 8309

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-of 6305 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-frec 6666 df-1o 6691 df-oadd 6695 df-er 6811 df-map 6928 df-pm 6929 df-en 7028 df-dom 7029 df-fin 7030 df-sup 7331 df-inf 7332 df-pnf 8369 df-mnf 8370 df-xr 8371 df-ltxr 8372 df-le 8373 df-sub 8507 df-neg 8508 df-reap 8912 df-ap 8919 df-div 9012 df-inn 9315 df-2 9373 df-3 9374 df-4 9375 df-5 9376 df-6 9377 df-7 9378 df-8 9379 df-9 9380 df-n0 9576 df-z 9657 df-dec 9790 df-uz 9939 df-q 10037 df-rp 10073 df-xneg 10192 df-xadd 10193 df-fz 10430 df-fzo 10568 df-seqfrec 10907 df-exp 10998 df-ihash 11238 df-cj 11630 df-re 11631 df-im 11632 df-rsqrt 11787 df-abs 11788 df-clim 12071 df-sumdc 12146 df-struct 13413 df-ndx 13414 df-slot 13415 df-base 13417 df-sets 13418 df-iress 13419 df-plusg 13504 df-mulr 13505 df-starv 13506 df-tset 13510 df-ple 13511 df-ds 13513 df-unif 13514 df-rest 13655 df-topn 13656 df-0g 13672 df-topgen 13674 df-mgm 13736 df-sgrp 13777 df-mnd 13790 df-grp 13868 df-minusg 13869 df-mulg 13983 df-subg 14033 df-cmn 14180 df-mgp 14326 df-ur 14371 df-ring 14410 df-cring 14411 df-subrg 14637 df-psmet 14990 df-xmet 14991 df-met 14992 df-bl 14993 df-mopn 14994 df-fg 14996 df-metu 14997 df-cnfld 15004 df-top 15236 df-topon 15249 df-topsp 15269 df-bases 15281 df-ntr 15334 df-cn 15426 df-cnp 15427 df-tx 15491 df-xms 15577 df-ms 15578 df-cncf 15809 df-limced 15894 df-dvap 15895 df-ply 15968

This theorem is used by: dvply2 16005

W3C validator