signsplypnf - Mathbox for Thierry Arnoux

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Theorem signsplypnf 34535

Description: The quotient of a polynomial 𝐹 by a monic monomial of same degree 𝐺 converges to the highest coefficient of 𝐹. (Contributed by Thierry Arnoux, 18-Sep-2018.)

**Hypotheses**
Ref	Expression
signsply0.d	⊢ 𝐷 = (deg‘𝐹)
signsply0.c	⊢ 𝐶 = (coeff‘𝐹)
signsply0.b	⊢ 𝐵 = (𝐶‘𝐷)
signsplypnf.g	⊢ 𝐺 = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))

**Assertion**
Ref	Expression
signsplypnf	⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) ⇝_𝑟 𝐵)

Distinct variable groups: 𝑥,𝐶 𝑥,𝐷 𝑥,𝐹 𝑥,𝐺 Allowed substitution hint: 𝐵(𝑥)

Proof of Theorem signsplypnf Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyf 26137	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
2	1	ffnd 6671	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹 Fn ℂ)
3		ovex 7402	. . . . . 6 ⊢ (𝑥↑𝐷) ∈ V
4	3	rgenw 3048	. . . . 5 ⊢ ∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V
5		signsplypnf.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))
6	5	fnmpt 6640	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V → 𝐺 Fn ℝ⁺)
7	4, 6	mp1i 13	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐺 Fn ℝ⁺)
8		cnex 11127	. . . . 5 ⊢ ℂ ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℂ ∈ V)
10		reex 11137	. . . . . 6 ⊢ ℝ ∈ V
11		rpssre 12937	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	10, 11	ssexi 5272	. . . . 5 ⊢ ℝ⁺ ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ∈ V)
14		ax-resscn 11103	. . . . . 6 ⊢ ℝ ⊆ ℂ
15	11, 14	sstri 3953	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		sseqin2 4182	. . . . 5 ⊢ (ℝ⁺ ⊆ ℂ ↔ (ℂ ∩ ℝ⁺) = ℝ⁺)
17	15, 16	mpbi 230	. . . 4 ⊢ (ℂ ∩ ℝ⁺) = ℝ⁺
18		signsply0.c	. . . . 5 ⊢ 𝐶 = (coeff‘𝐹)
19		signsply0.d	. . . . 5 ⊢ 𝐷 = (deg‘𝐹)
20	18, 19	coeid2 26178	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)))
21	5	fvmpt2 6961	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑥↑𝐷) ∈ V) → (𝐺‘𝑥) = (𝑥↑𝐷))
22	3, 21	mpan2 691	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (𝐺‘𝑥) = (𝑥↑𝐷))
23	22	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐺‘𝑥) = (𝑥↑𝐷))
24	2, 7, 9, 13, 17, 20, 23	offval 7642	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
25		fzfid 13916	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) ∈ Fin)
26	15	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ⊆ ℂ)
27	26	sselda 3943	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
28		dgrcl 26172	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → (deg‘𝐹) ∈ ℕ₀)
29	19, 28	eqeltrid 2832	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℕ₀)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
31	27, 30	expcld 14089	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
32	18	coef3 26171	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐶:ℕ₀⟶ℂ)
33	32	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐶:ℕ₀⟶ℂ)
34		elfznn0 13559	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐷) → 𝑘 ∈ ℕ₀)
35	34	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑘 ∈ ℕ₀)
36	33, 35	ffvelcdmd 7039	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝐶‘𝑘) ∈ ℂ)
37	27	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ∈ ℂ)
38	37, 35	expcld 14089	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝑘) ∈ ℂ)
39	36, 38	mulcld 11172	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → ((𝐶‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
40		rpne0 12946	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
41	40	adantl 481	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
42	29	nn0zd 12533	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℤ)
43	42	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
44	27, 41, 43	expne0d 14095	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
45	25, 31, 39, 44	fsumdivc 15729	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = Σ𝑘 ∈ (0...𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)))
46		fzosn 13675	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (𝐷..^(𝐷 + 1)) = {𝐷})
47	46	ineq2d 4179	. . . . . . . 8 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ((0..^𝐷) ∩ {𝐷}))
48		fzodisj 13632	. . . . . . . 8 ⊢ ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ∅
49	47, 48	eqtr3di 2779	. . . . . . 7 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ {𝐷}) = ∅)
50	43, 49	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((0..^𝐷) ∩ {𝐷}) = ∅)
51		fzval3 13673	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (0...𝐷) = (0..^(𝐷 + 1)))
52	42, 51	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = (0..^(𝐷 + 1)))
53		nn0uz 12813	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
54	29, 53	eleqtrdi 2838	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ (ℤ_≥‘0))
55		fzosplitsn 13714	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℤ_≥‘0) → (0..^(𝐷 + 1)) = ((0..^𝐷) ∪ {𝐷}))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^(𝐷 + 1)) = ((0..^𝐷) ∪ {𝐷}))
57	52, 56	eqtrd 2764	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
58	57	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
59	31	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ∈ ℂ)
60	41	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ≠ 0)
61	43	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐷 ∈ ℤ)
62	37, 60, 61	expne0d 14095	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ≠ 0)
63	39, 59, 62	divcld 11936	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ ℂ)
64	50, 58, 25, 63	fsumsplit 15684	. . . . 5 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (0...𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
65	45, 64	eqtrd 2764	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
66	65	mpteq2dva 5195	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)))))
67	24, 66	eqtrd 2764	. 2 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)))))
68		sumex 15631	. . . . 5 ⊢ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V
69	68	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V)
70		sumex 15631	. . . . 5 ⊢ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V
71	70	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V)
72	11	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ⊆ ℝ)
73		fzofi 13917	. . . . . . 7 ⊢ (0..^𝐷) ∈ Fin
74	73	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^𝐷) ∈ Fin)
75		ovexd 7404	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑘 ∈ (0..^𝐷))) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V)
76	32	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐶:ℕ₀⟶ℂ)
77		elfzonn0 13646	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 ∈ ℕ₀)
78	77	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℕ₀)
79	76, 78	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ ℂ)
80	27	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
81	80, 78	expcld 14089	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝑘) ∈ ℂ)
82	31	adantlr 715	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
83	40	adantl 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
84	43	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
85	80, 83, 84	expne0d 14095	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
86	79, 81, 82, 85	divassd 11971	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷))))
87	86	mpteq2dva 5195	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))))
88		fvexd 6855	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ V)
89		ovexd 7404	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑𝑘) / (𝑥↑𝐷)) ∈ V)
90	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐶:ℕ₀⟶ℂ)
91	77	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℕ₀)
92	90, 91	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐶‘𝑘) ∈ ℂ)
93		rlimconst 15487	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
94	11, 92, 93	sylancr 587	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
95	78	nn0zd 12533	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℤ)
96	84, 95	zsubcld 12621	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐷 − 𝑘) ∈ ℤ)
97	80, 83, 96	cxpexpzd 26654	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐷 − 𝑘)) = (𝑥↑(𝐷 − 𝑘)))
98	97	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (1 / (𝑥↑(𝐷 − 𝑘))))
99	80, 83, 96	expnegd 14096	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (1 / (𝑥↑(𝐷 − 𝑘))))
100	84	zcnd 12617	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℂ)
101	78	nn0cnd 12483	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
102	100, 101	negsubdi2d 11527	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → -(𝐷 − 𝑘) = (𝑘 − 𝐷))
103	102	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (𝑥↑(𝑘 − 𝐷)))
104	98, 99, 103	3eqtr2d 2770	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (𝑥↑(𝑘 − 𝐷)))
105	80, 83, 84, 95	expsubd 14100	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑(𝑘 − 𝐷)) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
106	104, 105	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
107	106	mpteq2dva 5195	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) = (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))))
108	91	nn0red 12482	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℝ)
109	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℕ₀)
110	109	nn0red 12482	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℝ)
111		elfzolt2 13607	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 < 𝐷)
112	111	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 < 𝐷)
113		difrp 12969	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑘 < 𝐷 ↔ (𝐷 − 𝑘) ∈ ℝ⁺))
114	113	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ 𝑘 < 𝐷) → (𝐷 − 𝑘) ∈ ℝ⁺)
115	108, 110, 112, 114	syl21anc 837	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐷 − 𝑘) ∈ ℝ⁺)
116		cxplim 26916	. . . . . . . . . . 11 ⊢ ((𝐷 − 𝑘) ∈ ℝ⁺ → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) ⇝_𝑟 0)
117	115, 116	syl 17	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) ⇝_𝑟 0)
118	107, 117	eqbrtrrd 5126	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))) ⇝_𝑟 0)
119	88, 89, 94, 118	rlimmul 15588	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 ((𝐶‘𝑘) · 0))
120	92	mul01d 11351	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → ((𝐶‘𝑘) · 0) = 0)
121	119, 120	breqtrd 5128	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 0)
122	87, 121	eqbrtrd 5124	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
123	72, 74, 75, 122	fsumrlim 15754	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 Σ𝑘 ∈ (0..^𝐷)0)
124	74	olcd 874	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → ((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin))
125		sumz 15665	. . . . . 6 ⊢ (((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin) → Σ𝑘 ∈ (0..^𝐷)0 = 0)
126	124, 125	syl 17	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → Σ𝑘 ∈ (0..^𝐷)0 = 0)
127	123, 126	breqtrd 5128	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
128	32, 29	ffvelcdmd 7039	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐶‘𝐷) ∈ ℂ)
129	128	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝐷) ∈ ℂ)
130	129, 31	mulcld 11172	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝐶‘𝐷) · (𝑥↑𝐷)) ∈ ℂ)
131	130, 31, 44	divcld 11936	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) ∈ ℂ)
132		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐶‘𝑘) = (𝐶‘𝐷))
133		oveq2 7377	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝑥↑𝑘) = (𝑥↑𝐷))
134	132, 133	oveq12d 7387	. . . . . . . . . 10 ⊢ (𝑘 = 𝐷 → ((𝐶‘𝑘) · (𝑥↑𝑘)) = ((𝐶‘𝐷) · (𝑥↑𝐷)))
135	134	oveq1d 7384	. . . . . . . . 9 ⊢ (𝑘 = 𝐷 → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
136	135	sumsn 15689	. . . . . . . 8 ⊢ ((𝐷 ∈ ℕ₀ ∧ (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) ∈ ℂ) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
137	30, 131, 136	syl2anc 584	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
138	129, 31, 44	divcan4d 11942	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
139	137, 138	eqtrd 2764	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
140	139	mpteq2dva 5195	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)))
141		rlimconst 15487	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝐷) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
142	11, 128, 141	sylancr 587	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
143	140, 142	eqbrtrd 5124	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 (𝐶‘𝐷))
144	69, 71, 127, 143	rlimadd 15586	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)))) ⇝_𝑟 (0 + (𝐶‘𝐷)))
145	128	addlidd 11353	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = (𝐶‘𝐷))
146		signsply0.b	. . . 4 ⊢ 𝐵 = (𝐶‘𝐷)
147	145, 146	eqtr4di 2782	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = 𝐵)
148	144, 147	breqtrd 5128	. 2 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) + Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)))) ⇝_𝑟 𝐵)
149	67, 148	eqbrtrd 5124	1 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) ⇝_𝑟 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 Vcvv 3444 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {csn 4585 class class class wbr 5102 ↦ cmpt 5183 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 Fincfn 8895 ℂcc 11044 ℝcr 11045 0cc0 11046 1c1 11047 + caddc 11049 · cmul 11051 < clt 11186 − cmin 11383 -cneg 11384 / cdiv 11813 ℕ₀cn0 12420 ℤcz 12507 ℤ_≥cuz 12771 ℝ⁺crp 12929 ...cfz 13446 ..^cfzo 13593 ↑cexp 14004 ⇝_𝑟 crli 15428 Σcsu 15629 Polycply 26123 coeffccoe 26125 degcdgr 26126 ↑_𝑐ccxp 26498

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9572 ax-cnex 11102 ax-resscn 11103 ax-1cn 11104 ax-icn 11105 ax-addcl 11106 ax-addrcl 11107 ax-mulcl 11108 ax-mulrcl 11109 ax-mulcom 11110 ax-addass 11111 ax-mulass 11112 ax-distr 11113 ax-i2m1 11114 ax-1ne0 11115 ax-1rid 11116 ax-rnegex 11117 ax-rrecex 11118 ax-cnre 11119 ax-pre-lttri 11120 ax-pre-lttrn 11121 ax-pre-ltadd 11122 ax-pre-mulgt0 11123 ax-pre-sup 11124 ax-addf 11125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9870 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11385 df-neg 11386 df-div 11814 df-nn 12165 df-2 12227 df-3 12228 df-4 12229 df-5 12230 df-6 12231 df-7 12232 df-8 12233 df-9 12234 df-n0 12421 df-z 12508 df-dec 12628 df-uz 12772 df-q 12886 df-rp 12930 df-xneg 13050 df-xadd 13051 df-xmul 13052 df-ioo 13288 df-ioc 13289 df-ico 13290 df-icc 13291 df-fz 13447 df-fzo 13594 df-fl 13732 df-mod 13810 df-seq 13945 df-exp 14005 df-fac 14217 df-bc 14246 df-hash 14274 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15414 df-clim 15431 df-rlim 15432 df-sum 15630 df-ef 16010 df-sin 16012 df-cos 16013 df-pi 16015 df-struct 17094 df-sets 17111 df-slot 17129 df-ndx 17141 df-base 17157 df-ress 17178 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17362 df-topn 17363 df-0g 17381 df-gsum 17382 df-topgen 17383 df-pt 17384 df-prds 17387 df-xrs 17442 df-qtop 17447 df-imas 17448 df-xps 17450 df-mre 17524 df-mrc 17525 df-acs 17527 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19232 df-cmn 19697 df-psmet 21289 df-xmet 21290 df-met 21291 df-bl 21292 df-mopn 21293 df-fbas 21294 df-fg 21295 df-cnfld 21298 df-top 22815 df-topon 22832 df-topsp 22854 df-bases 22867 df-cld 22940 df-ntr 22941 df-cls 22942 df-nei 23019 df-lp 23057 df-perf 23058 df-cn 23148 df-cnp 23149 df-haus 23236 df-tx 23483 df-hmeo 23676 df-fil 23767 df-fm 23859 df-flim 23860 df-flf 23861 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24805 df-0p 25605 df-limc 25801 df-dv 25802 df-ply 26127 df-coe 26129 df-dgr 26130 df-log 26499 df-cxp 26500

This theorem is referenced by: signsply0 34536

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