signsplypnf - Mathbox for Thierry Arnoux

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Theorem signsplypnf 34746

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 26185	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
2	1	ffnd 6660	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹 Fn ℂ)
3		ovex 7393	. . . . . 6 ⊢ (𝑥↑𝐷) ∈ V
4	3	rgenw 3059	. . . . 5 ⊢ ∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V
5		signsplypnf.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))
6	5	fnmpt 6629	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V → 𝐺 Fn ℝ⁺)
7	4, 6	mp1i 13	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐺 Fn ℝ⁺)
8		cnex 11114	. . . . 5 ⊢ ℂ ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℂ ∈ V)
10		reex 11124	. . . . . 6 ⊢ ℝ ∈ V
11		rpssre 12945	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	10, 11	ssexi 5253	. . . . 5 ⊢ ℝ⁺ ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ∈ V)
14		ax-resscn 11090	. . . . . 6 ⊢ ℝ ⊆ ℂ
15	11, 14	sstri 3926	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		sseqin2 4155	. . . . 5 ⊢ (ℝ⁺ ⊆ ℂ ↔ (ℂ ∩ ℝ⁺) = ℝ⁺)
17	15, 16	mpbi 232	. . . 4 ⊢ (ℂ ∩ ℝ⁺) = ℝ⁺
18		signsply0.c	. . . . 5 ⊢ 𝐶 = (coeff‘𝐹)
19		signsply0.d	. . . . 5 ⊢ 𝐷 = (deg‘𝐹)
20	18, 19	coeid2 26226	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)))
21	5	fvmpt2 6951	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑥↑𝐷) ∈ V) → (𝐺‘𝑥) = (𝑥↑𝐷))
22	3, 21	mpan2 698	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (𝐺‘𝑥) = (𝑥↑𝐷))
23	22	adantl 483	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐺‘𝑥) = (𝑥↑𝐷))
24	2, 7, 9, 13, 17, 20, 23	offval 7633	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
25		fzfid 13930	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) ∈ Fin)
26	15	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ⊆ ℂ)
27	26	sselda 3917	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
28		dgrcl 26220	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → (deg‘𝐹) ∈ ℕ₀)
29	19, 28	eqeltrid 2845	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℕ₀)
30	29	adantr 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
31	27, 30	expcld 14103	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
32	18	coef3 26219	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐶:ℕ₀⟶ℂ)
33	32	ad2antrr 733	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐶:ℕ₀⟶ℂ)
34		elfznn0 13569	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐷) → 𝑘 ∈ ℕ₀)
35	34	adantl 483	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑘 ∈ ℕ₀)
36	33, 35	ffvelcdmd 7030	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝐶‘𝑘) ∈ ℂ)
37	27	adantr 482	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ∈ ℂ)
38	37, 35	expcld 14103	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝑘) ∈ ℂ)
39	36, 38	mulcld 11160	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → ((𝐶‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
40		rpne0 12954	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
41	40	adantl 483	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
42	29	nn0zd 12544	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℤ)
43	42	adantr 482	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
44	27, 41, 43	expne0d 14109	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
45	25, 31, 39, 44	fsumdivc 15743	. . . . 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 13686	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (𝐷..^(𝐷 + 1)) = {𝐷})
47	46	ineq2d 4152	. . . . . . . 8 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ((0..^𝐷) ∩ {𝐷}))
48		fzodisj 13643	. . . . . . . 8 ⊢ ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ∅
49	47, 48	eqtr3di 2791	. . . . . . 7 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ {𝐷}) = ∅)
50	43, 49	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((0..^𝐷) ∩ {𝐷}) = ∅)
51		fzval3 13684	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (0...𝐷) = (0..^(𝐷 + 1)))
52	42, 51	syl 17	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = (0..^(𝐷 + 1)))
53		nn0uz 12821	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
54	29, 53	eleqtrdi 2851	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ (ℤ_≥‘0))
55		fzosplitsn 13726	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
58	57	adantr 482	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
59	31	adantr 482	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ∈ ℂ)
60	41	adantr 482	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ≠ 0)
61	43	adantr 482	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐷 ∈ ℤ)
62	37, 60, 61	expne0d 14109	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ≠ 0)
63	39, 59, 62	divcld 11926	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ ℂ)
64	50, 58, 25, 63	fsumsplit 15698	. . . . 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 2776	. . . 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 5168	. . 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 2776	. 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 15645	. . . . 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 15645	. . . . 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 13931	. . . . . . 7 ⊢ (0..^𝐷) ∈ Fin
74	73	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^𝐷) ∈ Fin)
75		ovexd 7395	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑘 ∈ (0..^𝐷))) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V)
76	32	ad2antrr 733	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐶:ℕ₀⟶ℂ)
77		elfzonn0 13657	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 ∈ ℕ₀)
78	77	ad2antlr 734	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℕ₀)
79	76, 78	ffvelcdmd 7030	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ ℂ)
80	27	adantlr 722	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
81	80, 78	expcld 14103	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝑘) ∈ ℂ)
82	31	adantlr 722	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
83	40	adantl 483	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
84	43	adantlr 722	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
85	80, 83, 84	expne0d 14109	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
86	79, 81, 82, 85	divassd 11961	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷))))
87	86	mpteq2dva 5168	. . . . . . 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 6846	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ V)
89		ovexd 7395	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑𝑘) / (𝑥↑𝐷)) ∈ V)
90	32	adantr 482	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐶:ℕ₀⟶ℂ)
91	77	adantl 483	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℕ₀)
92	90, 91	ffvelcdmd 7030	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐶‘𝑘) ∈ ℂ)
93		rlimconst 15501	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
94	11, 92, 93	sylancr 594	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
95	78	nn0zd 12544	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℤ)
96	84, 95	zsubcld 12633	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐷 − 𝑘) ∈ ℤ)
97	80, 83, 96	cxpexpzd 26697	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐷 − 𝑘)) = (𝑥↑(𝐷 − 𝑘)))
98	97	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (1 / (𝑥↑(𝐷 − 𝑘))))
99	80, 83, 96	expnegd 14110	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (1 / (𝑥↑(𝐷 − 𝑘))))
100	84	zcnd 12629	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℂ)
101	78	nn0cnd 12495	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
102	100, 101	negsubdi2d 11516	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → -(𝐷 − 𝑘) = (𝑘 − 𝐷))
103	102	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑-(𝐷 − 𝑘)) = (𝑥↑(𝑘 − 𝐷)))
104	98, 99, 103	3eqtr2d 2782	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (𝑥↑(𝑘 − 𝐷)))
105	80, 83, 84, 95	expsubd 14114	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑(𝑘 − 𝐷)) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
106	104, 105	eqtrd 2776	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
107	106	mpteq2dva 5168	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) = (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))))
108	91	nn0red 12494	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℝ)
109	29	adantr 482	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℕ₀)
110	109	nn0red 12494	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℝ)
111		elfzolt2 13618	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 < 𝐷)
112	111	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 < 𝐷)
113		difrp 12977	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑘 < 𝐷 ↔ (𝐷 − 𝑘) ∈ ℝ⁺))
114	113	biimpa 478	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ 𝑘 < 𝐷) → (𝐷 − 𝑘) ∈ ℝ⁺)
115	108, 110, 112, 114	syl21anc 844	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐷 − 𝑘) ∈ ℝ⁺)
116		cxplim 26957	. . . . . . . . . . 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 5099	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))) ⇝_𝑟 0)
119	88, 89, 94, 118	rlimmul 15602	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 ((𝐶‘𝑘) · 0))
120	92	mul01d 11340	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → ((𝐶‘𝑘) · 0) = 0)
121	119, 120	breqtrd 5101	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 0)
122	87, 121	eqbrtrd 5097	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
123	72, 74, 75, 122	fsumrlim 15769	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 Σ𝑘 ∈ (0..^𝐷)0)
124	74	olcd 881	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → ((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin))
125		sumz 15679	. . . . . 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 5101	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
128	32, 29	ffvelcdmd 7030	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐶‘𝐷) ∈ ℂ)
129	128	adantr 482	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝐷) ∈ ℂ)
130	129, 31	mulcld 11160	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝐶‘𝐷) · (𝑥↑𝐷)) ∈ ℂ)
131	130, 31, 44	divcld 11926	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) ∈ ℂ)
132		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐶‘𝑘) = (𝐶‘𝐷))
133		oveq2 7368	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝑥↑𝑘) = (𝑥↑𝐷))
134	132, 133	oveq12d 7378	. . . . . . . . . 10 ⊢ (𝑘 = 𝐷 → ((𝐶‘𝑘) · (𝑥↑𝑘)) = ((𝐶‘𝐷) · (𝑥↑𝐷)))
135	134	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑘 = 𝐷 → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
136	135	sumsn 15703	. . . . . . . 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 591	. . . . . . 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 11932	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
139	137, 138	eqtrd 2776	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
140	139	mpteq2dva 5168	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)))
141		rlimconst 15501	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝐷) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
142	11, 128, 141	sylancr 594	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
143	140, 142	eqbrtrd 5097	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 (𝐶‘𝐷))
144	69, 71, 127, 143	rlimadd 15600	. . 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 11342	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = (𝐶‘𝐷))
146		signsply0.b	. . . 4 ⊢ 𝐵 = (𝐶‘𝐷)
147	145, 146	eqtr4di 2794	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = 𝐵)
148	144, 147	breqtrd 5101	. 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 5097	1 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) ⇝_𝑟 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∨ wo 854 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 Vcvv 3433 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ∅c0 4264 {csn 4558 class class class wbr 5075 ↦ cmpt 5156 Fn wfn 6484 ⟶wf 6485 ‘cfv 6489 (class class class)co 7360 ∘_f cof 7622 Fincfn 8887 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 < clt 11174 − cmin 11372 -cneg 11373 / cdiv 11802 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 ℝ⁺crp 12937 ...cfz 13456 ..^cfzo 13603 ↑cexp 14018 ⇝_𝑟 crli 15442 Σcsu 15643 Polycply 26171 coeffccoe 26173 degcdgr 26174 ↑_𝑐ccxp 26541

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-mulg 19039 df-cntz 19287 df-cmn 19752 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-0p 25659 df-limc 25855 df-dv 25856 df-ply 26175 df-coe 26177 df-dgr 26178 df-log 26542 df-cxp 26543

This theorem is referenced by: signsply0 34747

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