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 26587

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 26543	. . . . 5 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ)
4	3	ffvelcdmda 7105	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
5	3	feqmptd 6973	. . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
6		plyco.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
7		eqid 2763	. . . . 5 ⊢ (coeff‘𝐹) = (coeff‘𝐹)
8		eqid 2763	. . . . 5 ⊢ (deg‘𝐹) = (deg‘𝐹)
9	7, 8	coeid 26584	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘))))
10	6, 9	syl 18	. . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘))))
11		oveq1 7447	. . . . 5 ⊢ (𝑥 = (𝐺‘𝑧) → (𝑥↑𝑘) = ((𝐺‘𝑧)↑𝑘))
12	11	oveq2d 7456	. . . 4 ⊢ (𝑥 = (𝐺‘𝑧) → (((coeff‘𝐹)‘𝑘) · (𝑥↑𝑘)) = (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
13	12	sumeq2sdv 15896	. . 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 7151	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))))
15		dgrcl 26579	. . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ₀)
16	6, 15	syl 18	. . 3 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ₀)
17		oveq2 7448	. . . . . . . 8 ⊢ (𝑥 = 0 → (0...𝑥) = (0...0))
18	17	sumeq1d 15893	. . . . . . 7 ⊢ (𝑥 = 0 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
19	18	mpteq2dv 5202	. . . . . 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 2849	. . . . 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 343	. . . 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 7448	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → (0...𝑥) = (0...𝑑))
23	22	sumeq1d 15893	. . . . . . 7 ⊢ (𝑥 = 𝑑 → Σ𝑘 ∈ (0...𝑥)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = Σ𝑘 ∈ (0...𝑑)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)))
24	23	mpteq2dv 5202	. . . . . 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 2849	. . . . 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 343	. . . 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 7448	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → (0...𝑥) = (0...(𝑑 + 1)))
28	27	sumeq1d 15893	. . . . . . 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 5202	. . . . . 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 2849	. . . . 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 343	. . . 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 7448	. . . . . . . 8 ⊢ (𝑥 = (deg‘𝐹) → (0...𝑥) = (0...(deg‘𝐹)))
33	32	sumeq1d 15893	. . . . . . 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 5202	. . . . . 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 2849	. . . . 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 343	. . . 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 12728	. . . . . . . . 9 ⊢ 0 ∈ ℤ
38	4	exp0d 14307	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑0) = 1)
39	38	oveq2d 7456	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = (((coeff‘𝐹)‘0) · 1))
40		plybss 26539	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
41	6, 40	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
42		0cnd 11323	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ∈ ℂ)
43	42	snssd 4747	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {0} ⊆ ℂ)
44	41, 43	unssd 4193	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ)
45	7	coef 26576	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
46	6, 45	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}))
47		0nn0 12645	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ ℕ₀
48		ffvelcdm 7102	. . . . . . . . . . . . . . 15 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ 0 ∈ ℕ₀) → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
49	46, 47, 48	sylancl 598	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0}))
50	44, 49	sseldd 3914	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((coeff‘𝐹)‘0) ∈ ℂ)
51	50	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘0) ∈ ℂ)
52	51	mulridd 11350	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · 1) = ((coeff‘𝐹)‘0))
53	39, 52	eqtrd 2798	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) = ((coeff‘𝐹)‘0))
54	53, 51	eqeltrd 2864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)) ∈ ℂ)
55		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘0))
56		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑0))
57	55, 56	oveq12d 7458	. . . . . . . . . 10 ⊢ (𝑘 = 0 → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
58	57	fsum1 15939	. . . . . . . . 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 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘0) · ((𝐺‘𝑧)↑0)))
60	59, 53	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = ((coeff‘𝐹)‘0))
61	60	mpteq2dva 5201	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0)))
62		fconstmpt 5717	. . . . . 6 ⊢ (ℂ × {((coeff‘𝐹)‘0)}) = (𝑧 ∈ ℂ ↦ ((coeff‘𝐹)‘0))
63	61, 62	eqtr4di 2816	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) = (ℂ × {((coeff‘𝐹)‘0)}))
64		plyconst 26551	. . . . . . 7 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘0) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
65	44, 49, 64	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘(𝑆 ∪ {0})))
66		plyun0 26542	. . . . . 6 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆)
67	65, 66	eleqtrdi 2874	. . . . 5 ⊢ (𝜑 → (ℂ × {((coeff‘𝐹)‘0)}) ∈ (Poly‘𝑆))
68	63, 67	eqeltrd 2864	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...0)(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
69		simprr 785	. . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑆 ∪ {0}) ⊆ ℂ)
71		peano2nn0 12670	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
72		ffvelcdm 7102	. . . . . . . . . . . . . 14 ⊢ (((coeff‘𝐹):ℕ₀⟶(𝑆 ∪ {0}) ∧ (𝑑 + 1) ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
73	46, 71, 72	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0}))
74		plyconst 26551	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∪ {0}) ⊆ ℂ ∧ ((coeff‘𝐹)‘(𝑑 + 1)) ∈ (𝑆 ∪ {0})) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
75	70, 73, 74	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘(𝑆 ∪ {0})))
76	75, 66	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∈ (Poly‘𝑆))
77		nn0p1nn 12669	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ)
78		oveq2 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 1 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑1))
79	78	mpteq2dv 5202	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 1 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)))
80	79	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 1 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆)))
81	80	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))))
82		oveq2 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑑 → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑𝑑))
83	82	mpteq2dv 5202	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
84	83	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆)))
85	84	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑑 → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))))
86		oveq2 7448	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑥) = ((𝐺‘𝑧)↑(𝑑 + 1)))
87	86	mpteq2dv 5202	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑑 + 1) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
88	87	eleq1d 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑑 + 1) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆) ↔ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
89	88	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑑 + 1) → ((𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑥)) ∈ (Poly‘𝑆)) ↔ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
90	4	exp1d 14308	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑1) = (𝐺‘𝑧))
91	90	mpteq2dva 5201	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
92	91, 5	eqtr4d 2801	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) = 𝐺)
93	92, 1	eqeltrd 2864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑1)) ∈ (Poly‘𝑆))
94		simprr 785	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))
95	1	adantr 486	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
100	94, 95, 97, 99	plymul 26564	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑑 ∈ ℕ ∧ (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆))) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆))
101	100	expr 462	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) ∈ (Poly‘𝑆)))
102		cnex 11305	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ ∈ V
103	102	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ℂ ∈ V)
104		ovexd 7475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑𝑑) ∈ V)
105	4	adantlr 728	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
106		eqidd 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)))
107	5	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → 𝐺 = (𝑧 ∈ ℂ ↦ (𝐺‘𝑧)))
108	103, 104, 105, 106, 107	offval2 7734	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
109		nnnn0 12637	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ∈ ℕ → 𝑑 ∈ ℕ₀)
110	109	ad2antlr 740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
111	105, 110	expp1d 14314	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) = (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧)))
112	111	mpteq2dva 5201	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ (((𝐺‘𝑧)↑𝑑) · (𝐺‘𝑧))))
113	108, 112	eqtr4d 2801	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ) → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∘_f · 𝐺) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
114	113	eleq1d 2849	. . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℕ → (𝜑 → ((𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑𝑑)) ∈ (Poly‘𝑆) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))))
117	116	a2d 30	. . . . . . . . . . . . . 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 12377	. . . . . . . . . . . . 13 ⊢ ((𝑑 + 1) ∈ ℕ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
119	77, 118	syl 18	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ₀ → (𝜑 → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆)))
120	119	impcom 413	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ (Poly‘𝑆))
121	96	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
122	98	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆)
123	76, 120, 121, 122	plymul 26564	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → ((ℂ × {((coeff‘𝐹)‘(𝑑 + 1))}) ∘_f · (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1)))) ∈ (Poly‘𝑆))
124	123	adantrr 730	. . . . . . . . 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 728	. . . . . . . . 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 26563	. . . . . . . 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 462	. . . . . . 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 15881	. . . . . . . . . . 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 7475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))) ∈ V)
132		eqidd 2764	. . . . . . . . . 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 6920	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((coeff‘𝐹)‘(𝑑 + 1)) ∈ V)
134		ovexd 7475	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → ((𝐺‘𝑧)↑(𝑑 + 1)) ∈ V)
135		fconstmpt 5717	. . . . . . . . . . . 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 2764	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑑 ∈ ℕ₀) → (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))) = (𝑧 ∈ ℂ ↦ ((𝐺‘𝑧)↑(𝑑 + 1))))
138	128, 133, 134, 136, 137	offval2 7734	. . . . . . . . . 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 7734	. . . . . . . . 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 781	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ ℕ₀)
141		nn0uz 13027	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
142	140, 141	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → 𝑑 ∈ (ℤ_≥‘0))
143	7	coef3 26578	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ₀⟶ℂ)
144	6, 143	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (coeff‘𝐹):ℕ₀⟶ℂ)
145	144	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (coeff‘𝐹):ℕ₀⟶ℂ)
146		elfznn0 13778	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0...(𝑑 + 1)) → 𝑘 ∈ ℕ₀)
147		ffvelcdm 7102	. . . . . . . . . . . . 13 ⊢ (((coeff‘𝐹):ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
148	145, 146, 147	syl2an 608	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((coeff‘𝐹)‘𝑘) ∈ ℂ)
149	4	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) → (𝐺‘𝑧) ∈ ℂ)
150		expcl 14246	. . . . . . . . . . . . 13 ⊢ (((𝐺‘𝑧) ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
151	149, 146, 150	syl2an 608	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → ((𝐺‘𝑧)↑𝑘) ∈ ℂ)
152	148, 151	mulcld 11353	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑑 ∈ ℕ₀) ∧ 𝑧 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑑 + 1))) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) ∈ ℂ)
153		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((coeff‘𝐹)‘𝑘) = ((coeff‘𝐹)‘(𝑑 + 1)))
154		oveq2 7448	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑑 + 1) → ((𝐺‘𝑧)↑𝑘) = ((𝐺‘𝑧)↑(𝑑 + 1)))
155	153, 154	oveq12d 7458	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑑 + 1) → (((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘)) = (((coeff‘𝐹)‘(𝑑 + 1)) · ((𝐺‘𝑧)↑(𝑑 + 1))))
156	142, 152, 155	fsump1 15948	. . . . . . . . . 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 5201	. . . . . . . . 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 2801	. . . . . . . 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 2849	. . . . . . 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 419	. . . . 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 30	. . . 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 12818	. . 3 ⊢ ((deg‘𝐹) ∈ ℕ₀ → (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆)))
164	16, 163	mpcom 39	. 2 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑘) · ((𝐺‘𝑧)↑𝑘))) ∈ (Poly‘𝑆))
165	14, 164	eqeltrd 2864	1 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (Poly‘𝑆))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 ⊆ wss 3900 ∪ cun 4152 {csn 4588 ↦ cmpt 5189 × cxp 5653 ∘ ccom 5659 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 ∘_f cof 7712 ℂcc 11222 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 ℕcn 12359 ℕ₀cn0 12630 ℤcz 12717 ℤ_≥cuz 12989 ...cfz 13663 ↑cexp 14228 Σcsu 15879 Polycply 26529 coeffccoe 26531 degcdgr 26532

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-inf2 9666 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301 ax-pre-sup 11302

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-of 7714 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-er 8742 df-map 8874 df-pm 8875 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-sup 9458 df-inf 9459 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-div 11998 df-nn 12360 df-2 12429 df-3 12430 df-n0 12631 df-z 12718 df-uz 12990 df-rp 13145 df-fz 13664 df-fzo 13813 df-fl 13956 df-seq 14169 df-exp 14229 df-hash 14499 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-clim 15681 df-rlim 15682 df-sum 15880 df-0p 26018 df-ply 26533 df-coe 26535 df-dgr 26536

This theorem is used by: dgrcolem1 26619 dgrcolem2 26620 preimaaa 26673 taylply2 26722 ftalem7 27433

W3C validator