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Theorem ftalem1 26566

Description: Lemma for fta 26573: "growth lemma". There exists some 𝑟 such that 𝐹 is arbitrarily close in proportion to its dominant term. (Contributed by Mario Carneiro, 14-Sep-2014.)

**Hypotheses**
Ref	Expression
ftalem.1	⊢ 𝐴 = (coeff‘𝐹)
ftalem.2	⊢ 𝑁 = (deg‘𝐹)
ftalem.3	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
ftalem.4	⊢ (𝜑 → 𝑁 ∈ ℕ)
ftalem1.5	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ftalem1.6	⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸)

**Assertion**
Ref	Expression
ftalem1	⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))

Distinct variable groups: 𝑘,𝑟,𝑥,𝐴 𝐸,𝑟 𝑘,𝑁,𝑟,𝑥 𝑘,𝐹,𝑟,𝑥 𝜑,𝑘,𝑥 𝑆,𝑘 𝑇,𝑘,𝑟,𝑥 Allowed substitution hints: 𝜑(𝑟) 𝑆(𝑥,𝑟) 𝐸(𝑥,𝑘)

**Proof of Theorem ftalem1**
Step	Hyp	Ref	Expression
1		ftalem1.6	. . . 4 ⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸)
2		fzfid 13934	. . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
4		ftalem.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
5	4	coef3 25737	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7		elfznn0 13590	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
8		ffvelcdm 7080	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
9	6, 7, 8	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
10	9	abscld 15379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
11	2, 10	fsumrecl 15676	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
12		ftalem1.5	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13	11, 12	rerpdivcld 13043	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) ∈ ℝ)
14	1, 13	eqeltrid 2837	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ)
15		1re 11210	. . 3 ⊢ 1 ∈ ℝ
16		ifcl 4572	. . 3 ⊢ ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
17	14, 15, 16	sylancl 586	. 2 ⊢ (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18		fzfid 13934	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
19	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ₀⟶ℂ)
20	19, 8	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
21		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
22		expcl 14041	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
23	21, 22	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
24	20, 23	mulcld 11230	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
25	7, 24	sylan2 593	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
26	18, 25	fsumcl 15675	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
27		ftalem.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
28	27	nnnn0d 12528	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
30	19, 29	ffvelcdmd 7084	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
31	21, 29	expcld 14107	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
32	30, 31	mulcld 11230	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
33	3	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
34		ftalem.2	. . . . . . . . . 10 ⊢ 𝑁 = (deg‘𝐹)
35	4, 34	coeid2 25744	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
36	33, 21, 35	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
37		nn0uz 12860	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
38	29, 37	eleqtrdi 2843	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘0))
39		elfznn0 13590	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
40	39, 24	sylan2 593	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
41		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝐴‘𝑘) = (𝐴‘𝑁))
42		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝑥↑𝑘) = (𝑥↑𝑁))
43	41, 42	oveq12d 7423	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((𝐴‘𝑘) · (𝑥↑𝑘)) = ((𝐴‘𝑁) · (𝑥↑𝑁)))
44	38, 40, 43	fsumm1 15693	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))))
45	36, 44	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))))
46	26, 32, 45	mvrraddd 11622	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)))
47	46	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))))
48	26	abscld 15379	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
49	25	abscld 15379	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
50	18, 49	fsumrecl 15676	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
51	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ⁺)
52	51	rpred 13012	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
53	21	abscld 15379	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
54	53, 29	reexpcld 14124	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
55	52, 54	remulcld 11240	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
56	18, 25	fsumabs 15743	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))))
57	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
58	27	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
59		nnm1nn0 12509	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
60	58, 59	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℕ₀)
61	53, 60	reexpcld 14124	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
62	57, 61	remulcld 11240	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
63	10	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
64	61	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
65	63, 64	remulcld 11240	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
66	20, 23	absmuld 15397	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) = ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))))
67	7, 66	sylan2 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) = ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))))
68	7, 23	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥↑𝑘) ∈ ℂ)
69	68	abscld 15379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ∈ ℝ)
70	7, 20	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
71	70	absge0d 15387	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴‘𝑘)))
72		absexp 15247	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
73	21, 7, 72	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
74	53	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘𝑥) ∈ ℝ)
75	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ∈ ℝ)
76	17	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
77		max1 13160	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
78	15, 14, 77	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
79	78	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
80		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))
81	75, 76, 53, 79, 80	lelttrd 11368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
82	75, 53, 81	ltled 11358	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
83	82	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
84		elfzuz3 13494	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
85	84	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
86	74, 83, 85	leexp2ad 14213	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
87	73, 86	eqbrtrd 5169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
88	69, 64, 63, 71, 87	lemul2ad 12150	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))) ≤ ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
89	67, 88	eqbrtrd 5169	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
90	18, 49, 65, 89	fsumle 15741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
91	61	recnd 11238	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
92	63	recnd 11238	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℂ)
93	18, 91, 92	fsummulc1 15727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
94	90, 93	breqtrrd 5175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
95	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
96		max2 13162	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
97	15, 14, 96	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
98	97	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
99	95, 76, 53, 98, 80	lelttrd 11368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
100	1, 99	eqbrtrrid 5183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) < (abs‘𝑥))
101	57, 53, 51	ltdivmuld 13063	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) < (abs‘𝑥) ↔ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥))))
102	100, 101	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)))
103	52, 53	remulcld 11240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
104	60	nn0zd 12580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
105		0red 11213	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
106		0lt1 11732	. . . . . . . . . . . . 13 ⊢ 0 < 1
107	106	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
108	105, 75, 53, 107, 81	lttrd 11371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
109		expgt0 14057	. . . . . . . . . . 11 ⊢ (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
110	53, 104, 108, 109	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
111		ltmul1 12060	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ ∧ (𝐸 · (abs‘𝑥)) ∈ ℝ ∧ (((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ ∧ 0 < ((abs‘𝑥)↑(𝑁 − 1)))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
112	57, 103, 61, 110, 111	syl112anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
113	102, 112	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
114	53	recnd 11238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
115		expm1t 14052	. . . . . . . . . . . 12 ⊢ (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
116	114, 58, 115	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
117	91, 114	mulcomd 11231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
118	116, 117	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
119	118	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
120	52	recnd 11238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
121	120, 114, 91	mulassd 11233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
122	119, 121	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
123	113, 122	breqtrrd 5175	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
124	50, 62, 55, 94, 123	lelttrd 11368	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
125	48, 50, 55, 56, 124	lelttrd 11368	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
126	47, 125	eqbrtrd 5169	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
127	126	expr 457	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
128	127	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
129		breq1 5150	. . 3 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
130	129	rspceaimv 3616	. 2 ⊢ ((if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ ∧ ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
131	17, 128, 130	syl2anc 584	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ifcif 4527 class class class wbr 5147 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ℝ⁺crp 12970 ...cfz 13480 ↑cexp 14023 abscabs 15177 Σcsu 15628 Polycply 25689 coeffccoe 25691 degcdgr 25692

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-0p 25178 df-ply 25693 df-coe 25695 df-dgr 25696

This theorem is referenced by: ftalem2 26567

W3C validator