signsplypnf - Mathbox for Thierry Arnoux

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Theorem signsplypnf 35246

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 26543	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹:ℂ⟶ℂ)
2	1	ffnd 6729	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐹 Fn ℂ)
3		ovex 7473	. . . . . 6 ⊢ (𝑥↑𝐷) ∈ V
4	3	rgenw 3084	. . . . 5 ⊢ ∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V
5		signsplypnf.g	. . . . . 6 ⊢ 𝐺 = (𝑥 ∈ ℝ⁺ ↦ (𝑥↑𝐷))
6	5	fnmpt 6698	. . . . 5 ⊢ (∀𝑥 ∈ ℝ⁺ (𝑥↑𝐷) ∈ V → 𝐺 Fn ℝ⁺)
7	4, 6	mp1i 14	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐺 Fn ℝ⁺)
8		cnex 11305	. . . . 5 ⊢ ℂ ∈ V
9	8	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℂ ∈ V)
10		reex 11315	. . . . . 6 ⊢ ℝ ∈ V
11		rpssre 13152	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	10, 11	ssexi 5288	. . . . 5 ⊢ ℝ⁺ ∈ V
13	12	a1i 11	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ∈ V)
14		ax-resscn 11281	. . . . . 6 ⊢ ℝ ⊆ ℂ
15	11, 14	sstri 3922	. . . . 5 ⊢ ℝ⁺ ⊆ ℂ
16		sseqin2 4123	. . . . 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 26585	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)))
21	5	fvmpt2 7025	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑥↑𝐷) ∈ V) → (𝐺‘𝑥) = (𝑥↑𝐷))
22	3, 21	mpan2 704	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (𝐺‘𝑥) = (𝑥↑𝐷))
23	22	adantl 487	. . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐺‘𝑥) = (𝑥↑𝐷))
24	2, 7, 9, 13, 17, 20, 23	offval 7723	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑘 ∈ (0...𝐷)((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))))
25		fzfid 14140	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) ∈ Fin)
26	15	a1i 11	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → ℝ⁺ ⊆ ℂ)
27	26	sselda 3913	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
28		dgrcl 26579	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → (deg‘𝐹) ∈ ℕ₀)
29	19, 28	eqeltrid 2868	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℕ₀)
30	29	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℕ₀)
31	27, 30	expcld 14313	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
32	18	coef3 26578	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐶:ℕ₀⟶ℂ)
33	32	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐶:ℕ₀⟶ℂ)
34		elfznn0 13778	. . . . . . . . 9 ⊢ (𝑘 ∈ (0...𝐷) → 𝑘 ∈ ℕ₀)
35	34	adantl 487	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑘 ∈ ℕ₀)
36	33, 35	ffvelcdmd 7106	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝐶‘𝑘) ∈ ℂ)
37	27	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ∈ ℂ)
38	37, 35	expcld 14313	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝑘) ∈ ℂ)
39	36, 38	mulcld 11353	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → ((𝐶‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
40		rpne0 13161	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
41	40	adantl 487	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
42	29	nn0zd 12742	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ ℤ)
43	42	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
44	27, 41, 43	expne0d 14319	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
45	25, 31, 39, 44	fsumdivc 15978	. . . . 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 13895	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (𝐷..^(𝐷 + 1)) = {𝐷})
47	46	ineq2d 4120	. . . . . . . 8 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ((0..^𝐷) ∩ {𝐷}))
48		fzodisj 13852	. . . . . . . 8 ⊢ ((0..^𝐷) ∩ (𝐷..^(𝐷 + 1))) = ∅
49	47, 48	eqtr3di 2813	. . . . . . 7 ⊢ (𝐷 ∈ ℤ → ((0..^𝐷) ∩ {𝐷}) = ∅)
50	43, 49	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((0..^𝐷) ∩ {𝐷}) = ∅)
51		fzval3 13893	. . . . . . . . 9 ⊢ (𝐷 ∈ ℤ → (0...𝐷) = (0..^(𝐷 + 1)))
52	42, 51	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = (0..^(𝐷 + 1)))
53		nn0uz 13027	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
54	29, 53	eleqtrdi 2874	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘ℝ) → 𝐷 ∈ (ℤ_≥‘0))
55		fzosplitsn 13935	. . . . . . . . 9 ⊢ (𝐷 ∈ (ℤ_≥‘0) → (0..^(𝐷 + 1)) = ((0..^𝐷) ∪ {𝐷}))
56	54, 55	syl 18	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^(𝐷 + 1)) = ((0..^𝐷) ∪ {𝐷}))
57	52, 56	eqtrd 2798	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
58	57	adantr 486	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (0...𝐷) = ((0..^𝐷) ∪ {𝐷}))
59	31	adantr 486	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ∈ ℂ)
60	41	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝑥 ≠ 0)
61	43	adantr 486	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → 𝐷 ∈ ℤ)
62	37, 60, 61	expne0d 14319	. . . . . . 7 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (𝑥↑𝐷) ≠ 0)
63	39, 59, 62	divcld 12117	. . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ (0...𝐷)) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ ℂ)
64	50, 58, 25, 63	fsumsplit 15933	. . . . 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 2798	. . . 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 5201	. . 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 2798	. 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 15881	. . . . 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 15881	. . . . 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 14141	. . . . . . 7 ⊢ (0..^𝐷) ∈ Fin
74	73	a1i 11	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0..^𝐷) ∈ Fin)
75		ovexd 7475	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ (𝑥 ∈ ℝ⁺ ∧ 𝑘 ∈ (0..^𝐷))) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) ∈ V)
76	32	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐶:ℕ₀⟶ℂ)
77		elfzonn0 13866	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 ∈ ℕ₀)
78	77	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℕ₀)
79	76, 78	ffvelcdmd 7106	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ ℂ)
80	27	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
81	80, 78	expcld 14313	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝑘) ∈ ℂ)
82	31	adantlr 728	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ∈ ℂ)
83	40	adantl 487	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
84	43	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℤ)
85	80, 83, 84	expne0d 14319	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑𝐷) ≠ 0)
86	79, 81, 82, 85	divassd 12152	. . . . . . . 8 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷))))
87	86	mpteq2dva 5201	. . . . . . 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 6920	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝑘) ∈ V)
89		ovexd 7475	. . . . . . . . 9 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → ((𝑥↑𝑘) / (𝑥↑𝐷)) ∈ V)
90	32	adantr 486	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐶:ℕ₀⟶ℂ)
91	77	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℕ₀)
92	90, 91	ffvelcdmd 7106	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐶‘𝑘) ∈ ℂ)
93		rlimconst 15737	. . . . . . . . . 10 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
94	11, 92, 93	sylancr 599	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝑘)) ⇝_𝑟 (𝐶‘𝑘))
95	78	nn0zd 12742	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℤ)
96	84, 95	zsubcld 12832	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝐷 − 𝑘) ∈ ℤ)
97	80, 83, 96	cxpexpzd 27066	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑_𝑐(𝐷 − 𝑘)) = (𝑥↑(𝐷 − 𝑘)))
98	97	oveq2d 7456	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (1 / (𝑥↑(𝐷 − 𝑘))))
99	80, 83, 96	expnegd 14320	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑ -(𝐷 − 𝑘)) = (1 / (𝑥↑(𝐷 − 𝑘))))
100	84	zcnd 12828	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝐷 ∈ ℂ)
101	78	nn0cnd 12693	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → 𝑘 ∈ ℂ)
102	100, 101	negsubdi2d 11709	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → -(𝐷 − 𝑘) = (𝑘 − 𝐷))
103	102	oveq2d 7456	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑ -(𝐷 − 𝑘)) = (𝑥↑(𝑘 − 𝐷)))
104	98, 99, 103	3eqtr2d 2804	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = (𝑥↑(𝑘 − 𝐷)))
105	80, 83, 84, 95	expsubd 14324	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (𝑥↑(𝑘 − 𝐷)) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
106	104, 105	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) ∧ 𝑥 ∈ ℝ⁺) → (1 / (𝑥↑_𝑐(𝐷 − 𝑘))) = ((𝑥↑𝑘) / (𝑥↑𝐷)))
107	106	mpteq2dva 5201	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) = (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))))
108	91	nn0red 12692	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 ∈ ℝ)
109	29	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℕ₀)
110	109	nn0red 12692	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝐷 ∈ ℝ)
111		elfzolt2 13827	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝐷) → 𝑘 < 𝐷)
112	111	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → 𝑘 < 𝐷)
113		difrp 13184	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝑘 < 𝐷 ↔ (𝐷 − 𝑘) ∈ ℝ⁺))
114	113	biimpa 482	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ℝ ∧ 𝐷 ∈ ℝ) ∧ 𝑘 < 𝐷) → (𝐷 − 𝑘) ∈ ℝ⁺)
115	108, 110, 112, 114	syl21anc 851	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝐷 − 𝑘) ∈ ℝ⁺)
116		cxplim 27326	. . . . . . . . . . 11 ⊢ ((𝐷 − 𝑘) ∈ ℝ⁺ → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) ⇝_𝑟 0)
117	115, 116	syl 18	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (1 / (𝑥↑_𝑐(𝐷 − 𝑘)))) ⇝_𝑟 0)
118	107, 117	eqbrtrrd 5132	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝑥↑𝑘) / (𝑥↑𝐷))) ⇝_𝑟 0)
119	88, 89, 94, 118	rlimmul 15838	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 ((𝐶‘𝑘) · 0))
120	92	mul01d 11533	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → ((𝐶‘𝑘) · 0) = 0)
121	119, 120	breqtrd 5134	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ ((𝐶‘𝑘) · ((𝑥↑𝑘) / (𝑥↑𝐷)))) ⇝_𝑟 0)
122	87, 121	eqbrtrd 5130	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑘 ∈ (0..^𝐷)) → (𝑥 ∈ ℝ⁺ ↦ (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
123	72, 74, 75, 122	fsumrlim 16004	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 Σ𝑘 ∈ (0..^𝐷)0)
124	74	olcd 888	. . . . . 6 ⊢ (𝐹 ∈ (Poly‘ℝ) → ((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin))
125		sumz 15914	. . . . . 6 ⊢ (((0..^𝐷) ⊆ (ℤ_≥‘0) ∨ (0..^𝐷) ∈ Fin) → Σ𝑘 ∈ (0..^𝐷)0 = 0)
126	124, 125	syl 18	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → Σ𝑘 ∈ (0..^𝐷)0 = 0)
127	123, 126	breqtrd 5134	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ (0..^𝐷)(((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 0)
128	32, 29	ffvelcdmd 7106	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐶‘𝐷) ∈ ℂ)
129	128	adantr 486	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (𝐶‘𝐷) ∈ ℂ)
130	129, 31	mulcld 11353	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → ((𝐶‘𝐷) · (𝑥↑𝐷)) ∈ ℂ)
131	130, 31, 44	divcld 12117	. . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) ∈ ℂ)
132		fveq2 6905	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝐶‘𝑘) = (𝐶‘𝐷))
133		oveq2 7448	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐷 → (𝑥↑𝑘) = (𝑥↑𝐷))
134	132, 133	oveq12d 7458	. . . . . . . . . 10 ⊢ (𝑘 = 𝐷 → ((𝐶‘𝑘) · (𝑥↑𝑘)) = ((𝐶‘𝐷) · (𝑥↑𝐷)))
135	134	oveq1d 7455	. . . . . . . . 9 ⊢ (𝑘 = 𝐷 → (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)))
136	135	sumsn 15938	. . . . . . . 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 596	. . . . . . 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 12123	. . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → (((𝐶‘𝐷) · (𝑥↑𝐷)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
139	137, 138	eqtrd 2798	. . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝑥 ∈ ℝ⁺) → Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷)) = (𝐶‘𝐷))
140	139	mpteq2dva 5201	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) = (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)))
141		rlimconst 15737	. . . . . 6 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (𝐶‘𝐷) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
142	11, 128, 141	sylancr 599	. . . . 5 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ (𝐶‘𝐷)) ⇝_𝑟 (𝐶‘𝐷))
143	140, 142	eqbrtrd 5130	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑘 ∈ {𝐷} (((𝐶‘𝑘) · (𝑥↑𝑘)) / (𝑥↑𝐷))) ⇝_𝑟 (𝐶‘𝐷))
144	69, 71, 127, 143	rlimadd 15836	. . 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 11535	. . . 4 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = (𝐶‘𝐷))
146		signsply0.b	. . . 4 ⊢ 𝐵 = (𝐶‘𝐷)
147	145, 146	eqtr4di 2816	. . 3 ⊢ (𝐹 ∈ (Poly‘ℝ) → (0 + (𝐶‘𝐷)) = 𝐵)
148	144, 147	breqtrd 5134	. 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 5130	1 ⊢ (𝐹 ∈ (Poly‘ℝ) → (𝐹 ∘_f / 𝐺) ⇝_𝑟 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 {csn 4588 class class class wbr 5106 ↦ cmpt 5189 Fn wfn 6550 ⟶wf 6551 ‘cfv 6555 (class class class)co 7440 ∘_f cof 7712 Fincfn 8998 ℂcc 11222 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 · cmul 11229 < clt 11367 − cmin 11565 -cneg 11566 / cdiv 11997 ℕ₀cn0 12630 ℤcz 12717 ℤ_≥cuz 12989 ℝ⁺crp 13144 ...cfz 13663 ..^cfzo 13812 ↑cexp 14228 ⇝_𝑟 crli 15678 Σcsu 15879 Polycply 26529 coeffccoe 26531 degcdgr 26532 ↑_𝑐ccxp 26910

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 ax-addf 11303

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-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 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-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-pm 8875 df-ixp 8951 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-fi 9427 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-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-q 13100 df-rp 13145 df-xneg 13265 df-xadd 13266 df-xmul 13267 df-ioo 13504 df-ioc 13505 df-ico 13506 df-icc 13507 df-fz 13664 df-fzo 13813 df-fl 13956 df-mod 14034 df-seq 14169 df-exp 14229 df-fac 14442 df-bc 14471 df-hash 14499 df-shft 15246 df-cj 15292 df-re 15293 df-im 15294 df-sqrt 15428 df-abs 15429 df-limsup 15664 df-clim 15681 df-rlim 15682 df-sum 15880 df-ef 16259 df-sin 16261 df-cos 16262 df-pi 16264 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-starv 17469 df-sca 17470 df-vsca 17471 df-ip 17472 df-tset 17473 df-ple 17474 df-ds 17476 df-unif 17477 df-hom 17478 df-cco 17479 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-pt 17641 df-prds 17644 df-xrs 17700 df-qtop 17705 df-imas 17706 df-xps 17708 df-mre 17782 df-mrc 17783 df-acs 17785 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-mulg 19304 df-cntz 19558 df-cmn 20023 df-psmet 21697 df-xmet 21698 df-met 21699 df-bl 21700 df-mopn 21701 df-fbas 21702 df-fg 21703 df-cnfld 21706 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-cld 23364 df-ntr 23365 df-cls 23366 df-nei 23443 df-lp 23481 df-perf 23482 df-cn 23572 df-cnp 23573 df-haus 23660 df-tx 23908 df-hmeo 24101 df-fil 24192 df-fm 24284 df-flim 24285 df-flf 24286 df-xms 24666 df-ms 24667 df-tms 24668 df-cncf 25226 df-0p 26018 df-limc 26213 df-dv 26214 df-ply 26533 df-coe 26535 df-dgr 26536 df-log 26911 df-cxp 26912

This theorem is used by: signsply0 35247

W3C validator