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Theorem plyco 26280

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 26237	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7104	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
5	3	feqmptd 6977	. . 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 26277	. . . 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 7438	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑧) → (𝑥↑𝑘) = ((𝐺‘𝑧)↑𝑘))
12	11	oveq2d 7447	. . . 4 ⊢ (𝑥 = (𝐺‘𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
13	12	sumeq2sdv 15739	. . 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 7149	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15		dgrcl 26272	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
16	6, 15	syl 17	. . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
17		oveq2 7439	. . . . . . . 8 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
18	17	sumeq1d 15736	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 5244	. . . . . 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 2826	. . . . 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 7439	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
23	22	sumeq1d 15736	. . . . . . 7 ⊢ (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
24	23	mpteq2dv 5244	. . . . . 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 2826	. . . . 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 7439	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
28	27	sumeq1d 15736	. . . . . . 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 5244	. . . . . 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 2826	. . . . 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 7439	. . . . . . . 8 ⊢ (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
33	32	sumeq1d 15736	. . . . . . 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 5244	. . . . . 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 2826	. . . . 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 12624	. . . . . . . . 9 ⊢ 0 ∈ ℤ
38	4	exp0d 14180	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
39	38	oveq2d 7447	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40		plybss 26233	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
42		0cnd 11254	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℂ)
43	42	snssd 4809	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {0} ⊆ ℂ)
44	41, 43	unssd 4192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
45	7	coef 26269	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
46	6, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
47		0nn0 12541	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ₀
48		ffvelcdm 7101	. . . . . . . . . . . . . . 15 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ₀) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
49	46, 47, 48	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
50	44, 49	sseldd 3984	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
52	51	mulridd 11278	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
53	39, 52	eqtrd 2777	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = ((coeff‘𝐹)‘0))
54	53, 51	eqeltrd 2841	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
55		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
57	55, 56	oveq12d 7449	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
58	57	fsum1 15783	. . . . . . . . 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 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
60	59, 53	eqtrd 2777	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
61	60	mpteq2dva 5242	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62		fconstmpt 5747	. . . . . 6 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
63	61, 62	eqtr4di 2795	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64		plyconst 26245	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
65	44, 49, 64	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66		plyun0 26236	. . . . . 6 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
67	65, 66	eleqtrdi 2851	. . . . 5 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
68	63, 67	eqeltrd 2841	. . . 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 12566	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
72		ffvelcdm 7101	. . . . . . . . . . . . . 14 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
73	46, 71, 72	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74		plyconst 26245	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
75	70, 73, 74	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
76	75, 66	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77		nn0p1nn 12565	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
78		oveq2 7439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑1))
79	78	mpteq2dv 5244	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
80	79	eleq1d 2826	. . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑𝑑))
83	82	mpteq2dv 5244	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
84	83	eleq1d 2826	. . . . . . . . . . . . . . 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 7439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑(𝑑 + 1)))
87	86	mpteq2dv 5244	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
88	87	eleq1d 2826	. . . . . . . . . . . . . . 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 14181	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
91	90	mpteq2dva 5242	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
92	91, 5	eqtr4d 2780	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
93	92, 1	eqeltrd 2841	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
100	94, 95, 97, 99	plymul 26257	. . . . . . . . . . . . . . . . . 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 11236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
104		ovexd 7466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑𝑑) ∈ V)
105	4	adantlr 715	. . . . . . . . . . . . . . . . . . . 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 7717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
109		nnnn0 12533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
110	109	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
111	105, 110	expp1d 14187	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
112	111	mpteq2dva 5242	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
113	108, 112	eqtr4d 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
114	113	eleq1d 2826	. . . . . . . . . . . . . . . . 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 12284	. . . . . . . . . . . . 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 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
122	98	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
123	76, 120, 121, 122	plymul 26257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124	123	adantrr 717	. . . . . . . . 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 715	. . . . . . . . 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 26256	. . . . . . . 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 15724	. . . . . . . . . . 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 7466	. . . . . . . . . 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 6921	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134		ovexd 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ V)
135		fconstmpt 5747	. . . . . . . . . . . 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 7717	. . . . . . . . . 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 7717	. . . . . . . . 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 12920	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
142	140, 141	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ_≥‘0))
143	7	coef3 26271	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶ℂ)
144	6, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶ℂ)
145	144	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ₀⟶ℂ)
146		elfznn0 13660	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ₀)
147		ffvelcdm 7101	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝐹):ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148	145, 146, 147	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
149	4	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
150		expcl 14120	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
151	149, 146, 150	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
152	148, 151	mulcld 11281	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
153		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
155	153, 154	oveq12d 7449	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
156	142, 152, 155	fsump1 15792	. . . . . . . . . 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 5242	. . . . . . . . 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 2780	. . . . . . . 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 2826	. . . . . . 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 12713	. . 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 2841	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 {csn 4626 ↦ cmpt 5225 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘_f cof 7695 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ℤ_≥cuz 12878 ...cfz 13547 ↑cexp 14102 Σcsu 15722 Polycply 26223 coeffccoe 26225 degcdgr 26226

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-0p 25705 df-ply 26227 df-coe 26229 df-dgr 26230

This theorem is referenced by: dgrcolem1 26313 dgrcolem2 26314 taylply2 26409 taylply2OLD 26410 ftalem7 27122

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