signsplypnf - Mathbox for Thierry Arnoux

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Theorem signsplypnf 32041

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 24887	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
2	1	ffnd 6500	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹 Fn ℂ)
3		ovex 7184	. . . . . 6 ⊢ (𝑥↑𝐷) ∈ V
4	3	rgenw 3083	. . . . 5 ⊢ ∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V
5		signsplypnf.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))
6	5	fnmpt 6472	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V → 𝐺 Fn ℝ⁺)
7	4, 6	mp1i 13	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐺 Fn ℝ⁺)
8		cnex 10649	. . . . 5 ⊢ ℂ ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℂ ∈ V)
10		reex 10659	. . . . . 6 ⊢ ℝ ∈ V
11		rpssre 12430	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	10, 11	ssexi 5193	. . . . 5 ⊢ ℝ⁺ ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ∈ V)
14		ax-resscn 10625	. . . . . 6 ⊢ ℝ ⊆ ℂ
15	11, 14	sstri 3902	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		sseqin2 4121	. . . . 5 ⊢ (ℝ⁺ ⊆ ℂ ↔ (ℂ ∩ ℝ⁺) = ℝ⁺)
17	15, 16	mpbi 233	. . . 4 ⊢ (ℂ ∩ ℝ⁺) = ℝ⁺
18		signsply0.c	. . . . 5 ⊢ 𝐶 = (coeff‘𝐹)
19		signsply0.d	. . . . 5 ⊢ 𝐷 = (deg‘𝐹)
20	18, 19	coeid2 24928	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)))
21	5	fvmpt2 6771	. . . . . 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 486	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐺‘𝑥) = (𝑥↑𝐷))
24	2, 7, 9, 13, 17, 20, 23	offval 7414	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
25		fzfid 13383	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) ∈ Fin)
26	15	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ⊆ ℂ)
27	26	sselda 3893	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
28		dgrcl 24922	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → (deg‘𝐹) ∈ ℕ₀)
29	19, 28	eqeltrid 2857	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℕ₀)
30	29	adantr 485	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
31	27, 30	expcld 13553	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
32	18	coef3 24921	. . . . . . . . 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 13042	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐷) → 𝑘 ∈ ℕ₀)
35	34	adantl 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑘 ∈ ℕ₀)
36	33, 35	ffvelrnd 6844	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝐶‘𝑘) ∈ ℂ)
37	27	adantr 485	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ∈ ℂ)
38	37, 35	expcld 13553	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝑘) ∈ ℂ)
39	36, 38	mulcld 10692	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → ((𝐶‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
40		rpne0 12439	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
41	40	adantl 486	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
42	29	nn0zd 12117	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℤ)
43	42	adantr 485	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
44	27, 41, 43	expne0d 13559	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
45	25, 31, 39, 44	fsumdivc 15182	. . . . 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		fzodisj 13113	. . . . . . . 8 ⊢ ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ∅
47		fzosn 13150	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (𝐷..^(𝐷 + 1)) = {𝐷})
48	47	ineq2d 4118	. . . . . . . 8 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ((0..^𝐷) ∩ {𝐷}))
49	46, 48	syl5reqr 2809	. . . . . . 7 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ {𝐷}) = ∅)
50	43, 49	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((0..^𝐷) ∩ {𝐷}) = ∅)
51		fzval3 13148	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (0...𝐷) = (0..^(𝐷 + 1)))
52	42, 51	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = (0..^(𝐷 + 1)))
53		nn0uz 12313	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
54	29, 53	eleqtrdi 2863	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ (ℤ_≥‘0))
55		fzosplitsn 13187	. . . . . . . . 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 2794	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
58	57	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
59	31	adantr 485	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ∈ ℂ)
60	41	adantr 485	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ≠ 0)
61	43	adantr 485	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐷 ∈ ℤ)
62	37, 60, 61	expne0d 13559	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ≠ 0)
63	39, 59, 62	divcld 11447	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ ℂ)
64	50, 58, 25, 63	fsumsplit 15138	. . . . 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 2794	. . . 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 5128	. . 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 2794	. 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 15085	. . . . 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 15085	. . . . 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 13384	. . . . . . 7 ⊢ (0..^𝐷) ∈ Fin
74	73	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^𝐷) ∈ Fin)
75		ovexd 7186	. . . . . 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 13124	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 ∈ ℕ₀)
78	77	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℕ₀)
79	76, 78	ffvelrnd 6844	. . . . . . . . 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 13553	. . . . . . . . 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 486	. . . . . . . . . 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 13559	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
86	79, 81, 82, 85	divassd 11482	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷))))
87	86	mpteq2dva 5128	. . . . . . 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 6674	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ V)
89		ovexd 7186	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑𝑘) / (𝑥↑𝐷)) ∈ V)
90	32	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐶:ℕ₀⟶ℂ)
91	77	adantl 486	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℕ₀)
92	90, 91	ffvelrnd 6844	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐶‘𝑘) ∈ ℂ)
93		rlimconst 14942	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
94	11, 92, 93	sylancr 591	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
95	78	nn0zd 12117	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℤ)
96	84, 95	zsubcld 12124	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐷 − 𝑘) ∈ ℤ)
97	80, 83, 96	cxpexpzd 25394	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐷 − 𝑘)) = (𝑥↑(𝐷 − 𝑘)))
98	97	oveq2d 7167	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (1 / (𝑥↑(𝐷 − 𝑘))))
99	80, 83, 96	expnegd 13560	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (1 / (𝑥↑(𝐷 − 𝑘))))
100	84	zcnd 12120	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℂ)
101	78	nn0cnd 11989	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
102	100, 101	negsubdi2d 11044	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → -(𝐷 − 𝑘) = (𝑘 − 𝐷))
103	102	oveq2d 7167	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (𝑥↑(𝑘 − 𝐷)))
104	98, 99, 103	3eqtr2d 2800	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (𝑥↑(𝑘 − 𝐷)))
105	80, 83, 84, 95	expsubd 13564	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑(𝑘 − 𝐷)) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
106	104, 105	eqtrd 2794	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
107	106	mpteq2dva 5128	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) = (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))))
108	91	nn0red 11988	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℝ)
109	29	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℕ₀)
110	109	nn0red 11988	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℝ)
111		elfzolt2 13089	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 < 𝐷)
112	111	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 < 𝐷)
113		difrp 12461	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑘 < 𝐷 ↔ (𝐷 − 𝑘) ∈ ℝ⁺))
114	113	biimpa 481	. . . . . . . . . . . 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 25649	. . . . . . . . . . 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 5057	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))) ⇝_𝑟 0)
119	88, 89, 94, 118	rlimmul 15042	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 ((𝐶‘𝑘) · 0))
120	92	mul01d 10870	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → ((𝐶‘𝑘) · 0) = 0)
121	119, 120	breqtrd 5059	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 0)
122	87, 121	eqbrtrd 5055	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
123	72, 74, 75, 122	fsumrlim 15207	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 Σ𝑘 ∈ (0..^𝐷)0)
124	74	olcd 872	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → ((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin))
125		sumz 15120	. . . . . 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 5059	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
128	32, 29	ffvelrnd 6844	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐶‘𝐷) ∈ ℂ)
129	128	adantr 485	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝐷) ∈ ℂ)
130	129, 31	mulcld 10692	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝐶‘𝐷) · (𝑥↑𝐷)) ∈ ℂ)
131	130, 31, 44	divcld 11447	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) ∈ ℂ)
132		fveq2 6659	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐶‘𝑘) = (𝐶‘𝐷))
133		oveq2 7159	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝑥↑𝑘) = (𝑥↑𝐷))
134	132, 133	oveq12d 7169	. . . . . . . . . 10 ⊢ (𝑘 = 𝐷 → ((𝐶‘𝑘) · (𝑥↑𝑘)) = ((𝐶‘𝐷) · (𝑥↑𝐷)))
135	134	oveq1d 7166	. . . . . . . . 9 ⊢ (𝑘 = 𝐷 → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
136	135	sumsn 15142	. . . . . . . 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 588	. . . . . . 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 11453	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
139	137, 138	eqtrd 2794	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
140	139	mpteq2dva 5128	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)))
141		rlimconst 14942	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝐷) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
142	11, 128, 141	sylancr 591	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
143	140, 142	eqbrtrd 5055	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 (𝐶‘𝐷))
144	69, 71, 127, 143	rlimadd 15040	. . 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	addid2d 10872	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = (𝐶‘𝐷))
146		signsply0.b	. . . 4 ⊢ 𝐵 = (𝐶‘𝐷)
147	145, 146	eqtr4di 2812	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = 𝐵)
148	144, 147	breqtrd 5059	. 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 5055	1 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) ⇝_𝑟 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∨ wo 845 = wceq 1539 ∈ wcel 2112 ≠ wne 2952 ∀wral 3071 Vcvv 3410 ∪ cun 3857 ∩ cin 3858 ⊆ wss 3859 ∅c0 4226 {csn 4523 class class class wbr 5033 ↦ cmpt 5113 Fn wfn 6331 ⟶wf 6332 ‘cfv 6336 (class class class)co 7151 ∘_f cof 7404 Fincfn 8528 ℂcc 10566 ℝcr 10567 0cc0 10568 1c1 10569 + caddc 10571 · cmul 10573 < clt 10706 − cmin 10901 -cneg 10902 / cdiv 11328 ℕ₀cn0 11927 ℤcz 12013 ℤ_≥cuz 12275 ℝ⁺crp 12423 ...cfz 12932 ..^cfzo 13075 ↑cexp 13472 ⇝_𝑟 crli 14883 Σcsu 15083 Polycply 24873 coeffccoe 24875 degcdgr 24876 ↑_𝑐ccxp 25239

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5157 ax-sep 5170 ax-nul 5177 ax-pow 5235 ax-pr 5299 ax-un 7460 ax-inf2 9130 ax-cnex 10624 ax-resscn 10625 ax-1cn 10626 ax-icn 10627 ax-addcl 10628 ax-addrcl 10629 ax-mulcl 10630 ax-mulrcl 10631 ax-mulcom 10632 ax-addass 10633 ax-mulass 10634 ax-distr 10635 ax-i2m1 10636 ax-1ne0 10637 ax-1rid 10638 ax-rnegex 10639 ax-rrecex 10640 ax-cnre 10641 ax-pre-lttri 10642 ax-pre-lttrn 10643 ax-pre-ltadd 10644 ax-pre-mulgt0 10645 ax-pre-sup 10646 ax-addf 10647 ax-mulf 10648

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-nel 3057 df-ral 3076 df-rex 3077 df-reu 3078 df-rmo 3079 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-pss 3878 df-nul 4227 df-if 4422 df-pw 4497 df-sn 4524 df-pr 4526 df-tp 4528 df-op 4530 df-uni 4800 df-int 4840 df-iun 4886 df-iin 4887 df-br 5034 df-opab 5096 df-mpt 5114 df-tr 5140 df-id 5431 df-eprel 5436 df-po 5444 df-so 5445 df-fr 5484 df-se 5485 df-we 5486 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6127 df-ord 6173 df-on 6174 df-lim 6175 df-suc 6176 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7109 df-ov 7154 df-oprab 7155 df-mpo 7156 df-of 7406 df-om 7581 df-1st 7694 df-2nd 7695 df-supp 7837 df-wrecs 7958 df-recs 8019 df-rdg 8057 df-1o 8113 df-2o 8114 df-oadd 8117 df-er 8300 df-map 8419 df-pm 8420 df-ixp 8481 df-en 8529 df-dom 8530 df-sdom 8531 df-fin 8532 df-fsupp 8860 df-fi 8901 df-sup 8932 df-inf 8933 df-oi 9000 df-card 9394 df-pnf 10708 df-mnf 10709 df-xr 10710 df-ltxr 10711 df-le 10712 df-sub 10903 df-neg 10904 df-div 11329 df-nn 11668 df-2 11730 df-3 11731 df-4 11732 df-5 11733 df-6 11734 df-7 11735 df-8 11736 df-9 11737 df-n0 11928 df-z 12014 df-dec 12131 df-uz 12276 df-q 12382 df-rp 12424 df-xneg 12541 df-xadd 12542 df-xmul 12543 df-ioo 12776 df-ioc 12777 df-ico 12778 df-icc 12779 df-fz 12933 df-fzo 13076 df-fl 13204 df-mod 13280 df-seq 13412 df-exp 13473 df-fac 13677 df-bc 13706 df-hash 13734 df-shft 14467 df-cj 14499 df-re 14500 df-im 14501 df-sqrt 14635 df-abs 14636 df-limsup 14869 df-clim 14886 df-rlim 14887 df-sum 15084 df-ef 15462 df-sin 15464 df-cos 15465 df-pi 15467 df-struct 16536 df-ndx 16537 df-slot 16538 df-base 16540 df-sets 16541 df-ress 16542 df-plusg 16629 df-mulr 16630 df-starv 16631 df-sca 16632 df-vsca 16633 df-ip 16634 df-tset 16635 df-ple 16636 df-ds 16638 df-unif 16639 df-hom 16640 df-cco 16641 df-rest 16747 df-topn 16748 df-0g 16766 df-gsum 16767 df-topgen 16768 df-pt 16769 df-prds 16772 df-xrs 16826 df-qtop 16831 df-imas 16832 df-xps 16834 df-mre 16908 df-mrc 16909 df-acs 16911 df-mgm 17911 df-sgrp 17960 df-mnd 17971 df-submnd 18016 df-mulg 18285 df-cntz 18507 df-cmn 18968 df-psmet 20151 df-xmet 20152 df-met 20153 df-bl 20154 df-mopn 20155 df-fbas 20156 df-fg 20157 df-cnfld 20160 df-top 21587 df-topon 21604 df-topsp 21626 df-bases 21639 df-cld 21712 df-ntr 21713 df-cls 21714 df-nei 21791 df-lp 21829 df-perf 21830 df-cn 21920 df-cnp 21921 df-haus 22008 df-tx 22255 df-hmeo 22448 df-fil 22539 df-fm 22631 df-flim 22632 df-flf 22633 df-xms 23015 df-ms 23016 df-tms 23017 df-cncf 23572 df-0p 24363 df-limc 24558 df-dv 24559 df-ply 24877 df-coe 24879 df-dgr 24880 df-log 25240 df-cxp 25241

This theorem is referenced by: signsply0 32042

W3C validator