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Theorem plyco 24838

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 24795	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelrnda 6828	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
5	3	feqmptd 6708	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
6		plyco.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
7		eqid 2798	. . . . 5 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
8		eqid 2798	. . . . 5 ⊢ (deg‘𝐹) = (deg‘𝐹)
9	7, 8	coeid 24835	. . . 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 7142	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑧) → (𝑥↑𝑘) = ((𝐺‘𝑧)↑𝑘))
12	11	oveq2d 7151	. . . 4 ⊢ (𝑥 = (𝐺‘𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
13	12	sumeq2sdv 15053	. . 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 6868	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15		dgrcl 24830	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
16	6, 15	syl 17	. . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
17		oveq2 7143	. . . . . . . 8 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
18	17	sumeq1d 15050	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 5126	. . . . . 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 2874	. . . . 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 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
23	22	sumeq1d 15050	. . . . . . 7 ⊢ (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
24	23	mpteq2dv 5126	. . . . . 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 2874	. . . . 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 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
28	27	sumeq1d 15050	. . . . . . 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 5126	. . . . . 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 2874	. . . . 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 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
33	32	sumeq1d 15050	. . . . . . 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 5126	. . . . . 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 2874	. . . . 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 344	. . . 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 11980	. . . . . . . . 9 ⊢ 0 ∈ ℤ
38	4	exp0d 13500	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
39	38	oveq2d 7151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40		plybss 24791	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
42		0cnd 10623	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℂ)
43	42	snssd 4702	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {0} ⊆ ℂ)
44	41, 43	unssd 4113	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
45	7	coef 24827	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
46	6, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
47		0nn0 11900	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ₀
48		ffvelrn 6826	. . . . . . . . . . . . . . 15 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ₀) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
49	46, 47, 48	sylancl 589	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
50	44, 49	sseldd 3916	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
51	50	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
52	51	mulid1d 10647	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
53	39, 52	eqtrd 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = ((coeff‘𝐹)‘0))
54	53, 51	eqeltrd 2890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
55		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
57	55, 56	oveq12d 7153	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
58	57	fsum1 15094	. . . . . . . . 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 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
60	59, 53	eqtrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
61	60	mpteq2dva 5125	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62		fconstmpt 5578	. . . . . 6 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
63	61, 62	eqtr4di 2851	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64		plyconst 24803	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
65	44, 49, 64	syl2anc 587	. . . . . 6 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66		plyun0 24794	. . . . . 6 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
67	65, 66	eleqtrdi 2900	. . . . 5 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
68	63, 67	eqeltrd 2890	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69		simprr 772	. . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑆 ∪ {0}) ⊆ ℂ)
71		peano2nn0 11925	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
72		ffvelrn 6826	. . . . . . . . . . . . . 14 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
73	46, 71, 72	syl2an 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74		plyconst 24803	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
75	70, 73, 74	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
76	75, 66	eleqtrdi 2900	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77		nn0p1nn 11924	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
78		oveq2 7143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑1))
79	78	mpteq2dv 5126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
80	79	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆)))
81	80	imbi2d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))))
82		oveq2 7143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑𝑑))
83	82	mpteq2dv 5126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
84	83	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
85	84	imbi2d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86		oveq2 7143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑(𝑑 + 1)))
87	86	mpteq2dv 5126	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
88	87	eleq1d 2874	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
89	88	imbi2d 344	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
90	4	exp1d 13501	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
91	90	mpteq2dva 5125	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
92	91, 5	eqtr4d 2836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
93	92, 1	eqeltrd 2890	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))
94		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))
95	1	adantr 484	. . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
100	94, 95, 97, 99	plymul 24815	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆))
101	100	expr 460	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆)))
102		cnex 10607	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
104		ovexd 7170	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑𝑑) ∈ V)
105	4	adantlr 714	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
106		eqidd 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
107	5	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
108	103, 104, 105, 106, 107	offval2 7406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
109		nnnn0 11892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
110	109	ad2antlr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
111	105, 110	expp1d 13507	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
112	111	mpteq2dva 5125	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
113	108, 112	eqtr4d 2836	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
114	113	eleq1d 2874	. . . . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
116	115	expcom 417	. . . . . . . . . . . . . . 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 11643	. . . . . . . . . . . . 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 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
121	96	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
122	98	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
123	76, 120, 121, 122	plymul 24815	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124	123	adantrr 716	. . . . . . . . 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 714	. . . . . . . . 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 24814	. . . . . . . 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 460	. . . . . . 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 15036	. . . . . . . . . . 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 7170	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ V)
132		eqidd 2799	. . . . . . . . . 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 6660	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134		ovexd 7170	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ V)
135		fconstmpt 5578	. . . . . . . . . . . 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 2799	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
138	128, 133, 134, 136, 137	offval2 7406	. . . . . . . . . 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 7406	. . . . . . . . 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 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
141		nn0uz 12268	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
142	140, 141	eleqtrdi 2900	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ_≥‘0))
143	7	coef3 24829	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶ℂ)
144	6, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶ℂ)
145	144	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ₀⟶ℂ)
146		elfznn0 12995	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ₀)
147		ffvelrn 6826	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝐹):ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148	145, 146, 147	syl2an 598	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
149	4	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
150		expcl 13443	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
151	149, 146, 150	syl2an 598	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
152	148, 151	mulcld 10650	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
153		fveq2 6645	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
155	153, 154	oveq12d 7153	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
156	142, 152, 155	fsump1 15103	. . . . . . . . . 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 5125	. . . . . . . . 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 2836	. . . . . . . 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 2874	. . . . . . 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 242	. . . . . 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 417	. . . . 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 12065	. . 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 2890	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 × cxp 5517 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘_f cof 7387 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 ↑cexp 13425 Σcsu 15034 Polycply 24781 coeffccoe 24783 degcdgr 24784

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788

This theorem is referenced by: dgrcolem1 24870 dgrcolem2 24871 taylply2 24963 ftalem7 25664

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