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Theorem plyco 26206

Description: The composition of two polynomials is a polynomial. (Contributed by Mario Carneiro, 23-Jul-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

**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)

**Assertion**
Ref	Expression
plyco	⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐺,𝑦 𝜑,𝑥,𝑦 𝑥,𝑆,𝑦

Proof of Theorem plyco Dummy variables 𝑘 𝑑 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyco.2	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆))
2		plyf 26163	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7031	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
5	3	feqmptd 6903	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
6		plyco.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
7		eqid 2737	. . . . 5 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
8		eqid 2737	. . . . 5 ⊢ (deg‘𝐹) = (deg‘𝐹)
9	7, 8	coeid 26203	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘))))
10	6, 9	syl 17	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘))))
11		oveq1 7367	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑧) → (𝑥↑𝑘) = ((𝐺‘𝑧)↑𝑘))
12	11	oveq2d 7376	. . . 4 ⊢ (𝑥 = (𝐺‘𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
13	12	sumeq2sdv 15630	. . 3 ⊢ (𝑥 = (𝐺‘𝑧) → Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
14	4, 5, 10, 13	fmptco 7076	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15		dgrcl 26198	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
16	6, 15	syl 17	. . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
17		oveq2 7368	. . . . . . . 8 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
18	17	sumeq1d 15627	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 5193	. . . . . 6 ⊢ (𝑥 = 0 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
20	19	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 0 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
21	20	imbi2d 340	. . . 4 ⊢ (𝑥 = 0 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
22		oveq2 7368	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
23	22	sumeq1d 15627	. . . . . . 7 ⊢ (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
24	23	mpteq2dv 5193	. . . . . 6 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
25	24	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
26	25	imbi2d 340	. . . 4 ⊢ (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
27		oveq2 7368	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
28	27	sumeq1d 15627	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
29	28	mpteq2dv 5193	. . . . . 6 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
30	29	eleq1d 2822	. . . . 5 ⊢ (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
31	30	imbi2d 340	. . . 4 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
32		oveq2 7368	. . . . . . . 8 ⊢ (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
33	32	sumeq1d 15627	. . . . . . 7 ⊢ (𝑥 = (deg‘𝐹) → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
34	33	mpteq2dv 5193	. . . . . 6 ⊢ (𝑥 = (deg‘𝐹) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
35	34	eleq1d 2822	. . . . 5 ⊢ (𝑥 = (deg‘𝐹) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
36	35	imbi2d 340	. . . 4 ⊢ (𝑥 = (deg‘𝐹) → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
37		0z 12503	. . . . . . . . 9 ⊢ 0 ∈ ℤ
38	4	exp0d 14067	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
39	38	oveq2d 7376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40		plybss 26159	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
42		0cnd 11129	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℂ)
43	42	snssd 4766	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {0} ⊆ ℂ)
44	41, 43	unssd 4145	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
45	7	coef 26195	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
46	6, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
47		0nn0 12420	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ₀
48		ffvelcdm 7028	. . . . . . . . . . . . . . 15 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ₀) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
49	46, 47, 48	sylancl 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
50	44, 49	sseldd 3935	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
52	51	mulridd 11153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
53	39, 52	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = ((coeff‘𝐹)‘0))
54	53, 51	eqeltrd 2837	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
55		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
57	55, 56	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
58	57	fsum1 15674	. . . . . . . . 9 ⊢ ((0 ∈ ℤ ∧ (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
59	37, 54, 58	sylancr 588	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
60	59, 53	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
61	60	mpteq2dva 5192	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62		fconstmpt 5687	. . . . . 6 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
63	61, 62	eqtr4di 2790	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64		plyconst 26171	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
65	44, 49, 64	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66		plyun0 26162	. . . . . 6 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
67	65, 66	eleqtrdi 2847	. . . . 5 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
68	63, 67	eqeltrd 2837	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
70	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑆 ∪ {0}) ⊆ ℂ)
71		peano2nn0 12445	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
72		ffvelcdm 7028	. . . . . . . . . . . . . 14 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
73	46, 71, 72	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74		plyconst 26171	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
75	70, 73, 74	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
76	75, 66	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77		nn0p1nn 12444	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
78		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑1))
79	78	mpteq2dv 5193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
80	79	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆)))
81	80	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))))
82		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑𝑑))
83	82	mpteq2dv 5193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
84	83	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
85	84	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑(𝑑 + 1)))
87	86	mpteq2dv 5193	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
88	87	eleq1d 2822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
89	88	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
90	4	exp1d 14068	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
91	90	mpteq2dva 5192	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
92	91, 5	eqtr4d 2775	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
93	92, 1	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))
94		simprr 773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))
95	1	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → 𝐺 ∈ (Poly‘𝑆))
96		plyco.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
97	96	adantlr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
98		plyco.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
99	98	adantlr 716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
100	94, 95, 97, 99	plymul 26183	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆))
101	100	expr 456	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆)))
102		cnex 11111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
104		ovexd 7395	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑𝑑) ∈ V)
105	4	adantlr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
106		eqidd 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
107	5	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
108	103, 104, 105, 106, 107	offval2 7644	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
109		nnnn0 12412	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
110	109	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
111	105, 110	expp1d 14074	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
112	111	mpteq2dva 5192	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
113	108, 112	eqtr4d 2775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
114	113	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
115	101, 114	sylibd 239	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116	115	expcom 413	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℕ → (𝜑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
117	116	a2d 29	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
118	81, 85, 89, 89, 93, 117	nnind 12167	. . . . . . . . . . . . 13 ⊢ ((𝑑 + 1) ∈ ℕ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
119	77, 118	syl 17	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
120	119	impcom 407	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
121	96	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
122	98	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
123	76, 120, 121, 122	plymul 26183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124	123	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
125	96	adantlr 716	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
126	69, 124, 125	plyadd 26182	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ₀ ∧ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘_f + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆))
127	126	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘_f + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆)))
128	102	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ℂ ∈ V)
129		sumex 15615	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ V
130	129	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ V)
131		ovexd 7395	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ V)
132		eqidd 2738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
133		fvexd 6850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134		ovexd 7395	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ V)
135		fconstmpt 5687	. . . . . . . . . . . 12 ⊢ (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1)))
136	135	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘(𝑑 + 1))))
137		eqidd 2738	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
138	128, 133, 134, 136, 137	offval2 7644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) = (𝑧 ∈ ℂ ↦ (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1)))))
139	128, 130, 131, 132, 138	offval2 7644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘_f + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))))
140		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
141		nn0uz 12793	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
142	140, 141	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ_≥‘0))
143	7	coef3 26197	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶ℂ)
144	6, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶ℂ)
145	144	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ₀⟶ℂ)
146		elfznn0 13540	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ₀)
147		ffvelcdm 7028	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝐹):ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148	145, 146, 147	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
149	4	adantlr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
150		expcl 14006	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
151	149, 146, 150	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
152	148, 151	mulcld 11156	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
153		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154		oveq2 7368	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
155	153, 154	oveq12d 7378	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
156	142, 152, 155	fsump1 15683	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1)))))
157	156	mpteq2dva 5192	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ (Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) + (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))))
158	139, 157	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘_f + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
159	158	eleq1d 2822	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∘_f + ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
160	127, 159	sylibd 239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
161	160	expcom 413	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → ((𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
162	161	a2d 29	. . . 4 ⊢ (𝑑 ∈ ℕ₀ → ((𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)) → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑑 + 1))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))))
163	21, 26, 31, 36, 68, 162	nn0ind 12591	. . 3 ⊢ ((deg‘𝐹) ∈ ℕ₀ → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
164	16, 163	mpcom 38	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
165	14, 164	eqeltrd 2837	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3441 ∪ cun 3900 ⊆ wss 3902 {csn 4581 ↦ cmpt 5180 × cxp 5623 ∘ ccom 5629 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ∘_f cof 7622 ℂcc 11028 0cc0 11030 1c1 11031 + caddc 11033 · cmul 11035 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ℤ_≥cuz 12755 ...cfz 13427 ↑cexp 13988 Σcsu 15613 Polycply 26149 coeffccoe 26151 degcdgr 26152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-n0 12406 df-z 12493 df-uz 12756 df-rp 12910 df-fz 13428 df-fzo 13575 df-fl 13716 df-seq 13929 df-exp 13989 df-hash 14258 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-clim 15415 df-rlim 15416 df-sum 15614 df-0p 25631 df-ply 26153 df-coe 26155 df-dgr 26156

This theorem is referenced by: dgrcolem1 26239 dgrcolem2 26240 taylply2 26335 taylply2OLD 26336 ftalem7 27049

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