plyco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > plyco		Structured version Visualization version GIF version

Theorem plyco 26204

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 26161	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7029	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
5	3	feqmptd 6901	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
6		plyco.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
7		eqid 2735	. . . . 5 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
8		eqid 2735	. . . . 5 ⊢ (deg‘𝐹) = (deg‘𝐹)
9	7, 8	coeid 26201	. . . 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 7365	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑧) → (𝑥↑𝑘) = ((𝐺‘𝑧)↑𝑘))
12	11	oveq2d 7374	. . . 4 ⊢ (𝑥 = (𝐺‘𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
13	12	sumeq2sdv 15628	. . 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 7074	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15		dgrcl 26196	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
16	6, 15	syl 17	. . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
17		oveq2 7366	. . . . . . . 8 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
18	17	sumeq1d 15625	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 5191	. . . . . 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 2820	. . . . 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 7366	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
23	22	sumeq1d 15625	. . . . . . 7 ⊢ (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
24	23	mpteq2dv 5191	. . . . . 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 2820	. . . . 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 7366	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
28	27	sumeq1d 15625	. . . . . . 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 5191	. . . . . 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 2820	. . . . 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 7366	. . . . . . . 8 ⊢ (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
33	32	sumeq1d 15625	. . . . . . 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 5191	. . . . . 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 2820	. . . . 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 12501	. . . . . . . . 9 ⊢ 0 ∈ ℤ
38	4	exp0d 14065	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
39	38	oveq2d 7374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40		plybss 26157	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
41	6, 40	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
42		0cnd 11127	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℂ)
43	42	snssd 4764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {0} ⊆ ℂ)
44	41, 43	unssd 4143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
45	7	coef 26193	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
46	6, 45	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
47		0nn0 12418	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ₀
48		ffvelcdm 7026	. . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
52	51	mulridd 11151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
53	39, 52	eqtrd 2770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = ((coeff‘𝐹)‘0))
54	53, 51	eqeltrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
55		fveq2 6833	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56		oveq2 7366	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
57	55, 56	oveq12d 7376	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
58	57	fsum1 15672	. . . . . . . . 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 2770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
61	60	mpteq2dva 5190	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62		fconstmpt 5685	. . . . . 6 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
63	61, 62	eqtr4di 2788	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64		plyconst 26169	. . . . . . 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 26160	. . . . . 6 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
67	65, 66	eleqtrdi 2845	. . . . 5 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
68	63, 67	eqeltrd 2835	. . . 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 12443	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
72		ffvelcdm 7026	. . . . . . . . . . . . . 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 26169	. . . . . . . . . . . . 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 2845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77		nn0p1nn 12442	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
78		oveq2 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑1))
79	78	mpteq2dv 5191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
80	79	eleq1d 2820	. . . . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑𝑑))
83	82	mpteq2dv 5191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
84	83	eleq1d 2820	. . . . . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑(𝑑 + 1)))
87	86	mpteq2dv 5191	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
88	87	eleq1d 2820	. . . . . . . . . . . . . . 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 14066	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
91	90	mpteq2dva 5190	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
92	91, 5	eqtr4d 2773	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
93	92, 1	eqeltrd 2835	. . . . . . . . . . . . . 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 26181	. . . . . . . . . . . . . . . . . 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 11109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
104		ovexd 7393	. . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . 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 7642	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
109		nnnn0 12410	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
110	109	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
111	105, 110	expp1d 14072	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
112	111	mpteq2dva 5190	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
113	108, 112	eqtr4d 2773	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
114	113	eleq1d 2820	. . . . . . . . . . . . . . . . 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 12165	. . . . . . . . . . . . 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 26181	. . . . . . . . . 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 26180	. . . . . . . 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 15613	. . . . . . . . . . 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 7393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ V)
132		eqidd 2736	. . . . . . . . . 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 6848	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134		ovexd 7393	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ V)
135		fconstmpt 5685	. . . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
138	128, 133, 134, 136, 137	offval2 7642	. . . . . . . . . 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 7642	. . . . . . . . 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 12791	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
142	140, 141	eleqtrdi 2845	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ_≥‘0))
143	7	coef3 26195	. . . . . . . . . . . . . . 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 13538	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ₀)
147		ffvelcdm 7026	. . . . . . . . . . . . 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 14004	. . . . . . . . . . . . 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 11154	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
153		fveq2 6833	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
155	153, 154	oveq12d 7376	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
156	142, 152, 155	fsump1 15681	. . . . . . . . . 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 5190	. . . . . . . . 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 2773	. . . . . . . 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 2820	. . . . . . 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 12589	. . 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 2835	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3439 ∪ cun 3898 ⊆ wss 3900 {csn 4579 ↦ cmpt 5178 × cxp 5621 ∘ ccom 5627 ⟶wf 6487 ‘cfv 6491 (class class class)co 7358 ∘_f cof 7620 ℂcc 11026 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 ℕcn 12147 ℕ₀cn0 12403 ℤcz 12490 ℤ_≥cuz 12753 ...cfz 13425 ↑cexp 13986 Σcsu 15611 Polycply 26147 coeffccoe 26149 degcdgr 26150

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 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-inf2 9552 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4902 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-isom 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-pm 8768 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-sup 9347 df-inf 9348 df-oi 9417 df-card 9853 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-rp 12908 df-fz 13426 df-fzo 13573 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612 df-0p 25629 df-ply 26151 df-coe 26153 df-dgr 26154

This theorem is referenced by: dgrcolem1 26237 dgrcolem2 26238 taylply2 26333 taylply2OLD 26334 ftalem7 27047

W3C validator