Theorem plycolemc 15274

Description: Lemma for plyco 15275. The result expressed as a sum, with a degree and coefficients for 𝐹 specified as hypotheses. (Contributed by Jim Kingdon, 20-Sep-2025.)

Hypotheses

Ref	Expression
plyco.1	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
plyco.2	⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
plyco.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
plyco.4	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
plycolemc.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
plycolemc.a	⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
plycolemc.z	⊢ (𝜑 → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0})
plycolemc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘))))

Assertion

Ref	Expression
plycolemc	⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))

Distinct variable groups:   𝑘,𝐺,𝑧   𝐴,𝑘   𝑘,𝑁   𝑥,𝐴,𝑦,𝑧,𝑘   𝑥,𝐺,𝑦   𝑧,𝑁   𝑥,𝑆,𝑦   𝜑,𝑘,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑆(𝑧,𝑘)   𝐹(𝑥,𝑦,𝑧,𝑘)   𝑁(𝑥,𝑦)

Proof of Theorem plycolemc

Dummy variables 𝑑 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plycolemc.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
2		oveq2 5959	. . . . . . 7 ⊢ (𝑤 = 0 → (0...𝑤) = (0...0))
3	2	sumeq1d 11721	. . . . . 6 ⊢ (𝑤 = 0 → Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
4	3	mpteq2dv 4139	. . . . 5 ⊢ (𝑤 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
5	4	eleq1d 2275	. . . 4 ⊢ (𝑤 = 0 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
6	5	imbi2d 230	. . 3 ⊢ (𝑤 = 0 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
7		oveq2 5959	. . . . . . 7 ⊢ (𝑤 = 𝑑 → (0...𝑤) = (0...𝑑))
8	7	sumeq1d 11721	. . . . . 6 ⊢ (𝑤 = 𝑑 → Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
9	8	mpteq2dv 4139	. . . . 5 ⊢ (𝑤 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
10	9	eleq1d 2275	. . . 4 ⊢ (𝑤 = 𝑑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
11	10	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
12		oveq2 5959	. . . . . . 7 ⊢ (𝑤 = (𝑑 + 1) → (0...𝑤) = (0...(𝑑 + 1)))
13	12	sumeq1d 11721	. . . . . 6 ⊢ (𝑤 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
14	13	mpteq2dv 4139	. . . . 5 ⊢ (𝑤 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15	14	eleq1d 2275	. . . 4 ⊢ (𝑤 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
16	15	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
17		oveq2 5959	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (0...𝑤) = (0...𝑁))
18	17	sumeq1d 11721	. . . . . 6 ⊢ (𝑤 = 𝑁 → Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 4139	. . . . 5 ⊢ (𝑤 = 𝑁 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
20	19	eleq1d 2275	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
21	20	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑤)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
22		0z 9390	. . . . . . . 8 ⊢ 0 ∈ ℤ
23		plyco.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
24		plyf 15253	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
25	23, 24	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
26	25	ffvelcdmda 5722	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
27	26	exp0d 10819	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
28	27	oveq2d 5967	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐴‘0) · ((𝐺‘𝑧)↑0)) = ((𝐴‘0) · 1))
29		plyco.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
30		plybss 15249	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
31	29, 30	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
32		0cnd 8072	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℂ)
33	32	snssd 3780	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {0} ⊆ ℂ)
34	31, 33	unssd 3350	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
35		plycolemc.a	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
36		0nn0 9317	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ0
37	36	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℕ0)
38	35, 37	ffvelcdmd 5723	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴‘0) ∈ (𝑆 ∪ {0}))
39	34, 38	sseldd 3195	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴‘0) ∈ ℂ)
40	39	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐴‘0) ∈ ℂ)
41	40	mulridd 8096	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐴‘0) · 1) = (𝐴‘0))
42	28, 41	eqtrd 2239	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐴‘0) · ((𝐺‘𝑧)↑0)) = (𝐴‘0))
43	42, 40	eqeltrd 2283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐴‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
44		fveq2 5583	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (𝐴‘𝑘) = (𝐴‘0))
45		oveq2 5959	. . . . . . . . . 10 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
46	44, 45	oveq12d 5969	. . . . . . . . 9 ⊢ (𝑘 = 0 → ((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((𝐴‘0) · ((𝐺‘𝑧)↑0)))
47	46	fsum1 11767	. . . . . . . 8 ⊢ ((0 ∈ ℤ ∧ ((𝐴‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((𝐴‘0) · ((𝐺‘𝑧)↑0)))
48	22, 43, 47	sylancr 414	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((𝐴‘0) · ((𝐺‘𝑧)↑0)))
49	48, 42	eqtrd 2239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (𝐴‘0))
50	49	mpteq2dva 4138	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ (𝐴‘0)))
51		fconstmpt 4726	. . . . 5 ⊢ (ℂ × {(𝐴‘0)}) = (𝑧 ∈ ℂ ↦ (𝐴‘0))
52	50, 51	eqtr4di 2257	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {(𝐴‘0)}))
53		plyconst 15261	. . . . . 6 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ (𝐴‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {(𝐴‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
54	34, 38, 53	syl2anc 411	. . . . 5 ⊢ (𝜑 → (ℂ × {(𝐴‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
55		plyun0 15252	. . . . 5 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
56	54, 55	eleqtrdi 2299	. . . 4 ⊢ (𝜑 → (ℂ × {(𝐴‘0)}) ∈ (Poly‘𝑆))
57	52, 56	eqeltrd 2283	. . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
58		simprr 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
59	34	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝑆 ∪ {0}) ⊆ ℂ)
60		peano2nn0 9342	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
61		ffvelcdm 5720	. . . . . . . . . . . . 13 ⊢ ((𝐴:ℕ0⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ0) → (𝐴‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
62	35, 60, 61	syl2an 289	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝐴‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
63		plyconst 15261	. . . . . . . . . . . 12 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ (𝐴‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {(𝐴‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
64	59, 62, 63	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (ℂ × {(𝐴‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
65	64, 55	eleqtrdi 2299	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (ℂ × {(𝐴‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
66		nn0p1nn 9341	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ)
67		oveq2 5959	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 1 → ((𝐺‘𝑧)↑𝑤) = ((𝐺‘𝑧)↑1))
68	67	mpteq2dv 4139	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
69	68	eleq1d 2275	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆)))
70	69	imbi2d 230	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))))
71		oveq2 5959	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑑 → ((𝐺‘𝑧)↑𝑤) = ((𝐺‘𝑧)↑𝑑))
72	71	mpteq2dv 4139	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
73	72	eleq1d 2275	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
74	73	imbi2d 230	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
75		oveq2 5959	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑤) = ((𝐺‘𝑧)↑(𝑑 + 1)))
76	75	mpteq2dv 4139	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
77	76	eleq1d 2275	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
78	77	imbi2d 230	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑤)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
79	26	exp1d 10820	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
80	79	mpteq2dva 4138	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
81	25	feqmptd 5639	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
82	80, 81	eqtr4d 2242	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
83	82, 23	eqeltrd 2283	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))
84		simprr 531	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))
85	23	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
86		plyco.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
87	86	adantlr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
88		plyco.4	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
89	88	adantlr 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
90	84, 85, 87, 89	plymul 15268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆))
91	90	expr 375	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆)))
92		cnex 8056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℂ ∈ V
93	92	a1i 9	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
94	26	adantlr 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
95		nnnn0 9309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ0)
96	95	ad2antlr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
97	94, 96	expcld 10825	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑𝑑) ∈ ℂ)
98	25	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺:ℂ⟶ℂ)
99	98	ffvelcdmda 5722	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
100		eqidd 2207	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
101	81	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
102	93, 97, 99, 100, 101	offval2 6181	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
103	94, 96	expp1d 10826	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
104	103	mpteq2dva 4138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
105	102, 104	eqtr4d 2242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘𝑓 · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
106	105	eleq1d 2275	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
107	91, 106	sylibd 149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
108	107	expcom 116	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → (𝜑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
109	108	a2d 26	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
110	70, 74, 78, 78, 83, 109	nnind 9059	. . . . . . . . . . . 12 ⊢ ((𝑑 + 1) ∈ ℕ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
111	66, 110	syl 14	. . . . . . . . . . 11 ⊢ (𝑑 ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
112	111	impcom 125	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
113	86	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
114	88	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
115	65, 112, 113, 114	plymul 15268	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
116	115	adantrr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
117	86	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
118	58, 116, 117	plyadd 15267	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ0 ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆))
119	118	expr 375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆)))
120	92	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ℂ ∈ V)
121		0zd 9391	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 0 ∈ ℤ)
122		simplr 528	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ0)
123	122	nn0zd 9500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℤ)
124	121, 123	fzfigd 10583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (0...𝑑) ∈ Fin)
125	29, 55	eleqtrrdi 2300	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (Poly‘(𝑆 ∪ {0})))
126		plybss 15249	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (Poly‘(𝑆 ∪ {0})) → (𝑆 ∪ {0}) ⊆ ℂ)
127	125, 126	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
128	127	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → (𝑆 ∪ {0}) ⊆ ℂ)
129	35	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
130		elfznn0 10243	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...𝑑) → 𝑘 ∈ ℕ0)
131	130	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → 𝑘 ∈ ℕ0)
132	129, 131	ffvelcdmd 5723	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → (𝐴‘𝑘) ∈ (𝑆 ∪ {0}))
133	128, 132	sseldd 3195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → (𝐴‘𝑘) ∈ ℂ)
134	26	ad4ant13 513	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → (𝐺‘𝑧) ∈ ℂ)
135	134, 131	expcld 10825	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
136	133, 135	mulcld 8100	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑑)) → ((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
137	124, 136	fsumcl 11755	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
138	127	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑆 ∪ {0}) ⊆ ℂ)
139	62	adantr 276	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐴‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
140	138, 139	sseldd 3195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐴‘(𝑑 + 1)) ∈ ℂ)
141	26	adantlr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
142	60	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → (𝑑 + 1) ∈ ℕ0)
143	141, 142	expcld 10825	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ ℂ)
144	140, 143	mulcld 8100	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ ℂ)
145		eqidd 2207	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
146		fconstmpt 4726	. . . . . . . . . . 11 ⊢ (ℂ × {(𝐴‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ (𝐴‘(𝑑 + 1)))
147	146	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (ℂ × {(𝐴‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ (𝐴‘(𝑑 + 1))))
148		eqidd 2207	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
149	120, 139, 143, 147, 148	offval2 6181	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) = (𝑧 ∈ ℂ ↦ ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1)))))
150	120, 137, 144, 145, 149	offval2 6181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))))
151		nn0uz 9690	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
152	122, 151	eleqtrdi 2299	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ≥‘0))
153	138	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (𝑆 ∪ {0}) ⊆ ℂ)
154	35	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → 𝐴:ℕ0⟶(𝑆 ∪ {0}))
155		elfznn0 10243	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ0)
156	155	adantl 277	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → 𝑘 ∈ ℕ0)
157	154, 156	ffvelcdmd 5723	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (𝐴‘𝑘) ∈ (𝑆 ∪ {0}))
158	153, 157	sseldd 3195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (𝐴‘𝑘) ∈ ℂ)
159	141	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (𝐺‘𝑧) ∈ ℂ)
160	159, 156	expcld 10825	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
161	158, 160	mulcld 8100	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
162		fveq2 5583	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (𝐴‘𝑘) = (𝐴‘(𝑑 + 1)))
163		oveq2 5959	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
164	162, 163	oveq12d 5969	. . . . . . . . . 10 ⊢ (𝑘 = (𝑑 + 1) → ((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
165	152, 161, 164	fsump1 11775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ0) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1)))))
166	165	mpteq2dva 4138	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + ((𝐴‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))))
167	150, 166	eqtr4d 2242	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
168	167	eleq1d 2275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → (((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘𝑓 + ((ℂ × {(𝐴‘(𝑑 + 1))}) ∘𝑓 · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
169	119, 168	sylibd 149	. . . . 5 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ0) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
170	169	expcom 116	. . . 4 ⊢ (𝑑 ∈ ℕ0 → (𝜑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
171	170	a2d 26	. . 3 ⊢ (𝑑 ∈ ℕ0 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
172	6, 11, 16, 21, 57, 171	nn0ind 9494	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
173	1, 172	mpcom 36	1 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∪ cun 3165   ⊆ wss 3167  {csn 3634   ↦ cmpt 4109   × cxp 4677   “ cima 4682  ⟶wf 5272  ‘cfv 5276  (class class class)co 5951   ∘_𝑓 cof 6163  ℂcc 7930  0cc0 7932  1c1 7933   + caddc 7935   · cmul 7937  ℕcn 9043  ℕ₀cn0 9302  ℤcz 9379  ℤ_≥cuz 9655  ...cfz 10137  ↑cexp 10690  Σcsu 11708  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

This theorem depends on definitions:  df-bi 117  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-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-disj 4024  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-en 6835  df-dom 6836  df-fin 6837  df-sup 7093  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-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  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-ply 15246

This theorem is referenced by:  plyco  15275