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Theorem ftalem1 26127

Description: Lemma for fta 26134: "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 13621	. . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
4		ftalem.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
5	4	coef3 25298	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7		elfznn0 13278	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
8		ffvelrn 6941	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
9	6, 7, 8	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
10	9	abscld 15076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
11	2, 10	fsumrecl 15374	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
12		ftalem1.5	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13	11, 12	rerpdivcld 12732	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) ∈ ℝ)
14	1, 13	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ)
15		1re 10906	. . 3 ⊢ 1 ∈ ℝ
16		ifcl 4501	. . 3 ⊢ ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
17	14, 15, 16	sylancl 585	. 2 ⊢ (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18		fzfid 13621	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
19	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ₀⟶ℂ)
20	19, 8	sylan 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
21		simprl 767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
22		expcl 13728	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
23	21, 22	sylan 579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
24	20, 23	mulcld 10926	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
25	7, 24	sylan2 592	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
26	18, 25	fsumcl 15373	. . . . . . 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 12223	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
30	19, 29	ffvelrnd 6944	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
31	21, 29	expcld 13792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
32	30, 31	mulcld 10926	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
33	3	adantr 480	. . . . . . . . 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 25305	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
36	33, 21, 35	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
37		nn0uz 12549	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
38	29, 37	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘0))
39		elfznn0 13278	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
40	39, 24	sylan2 592	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
41		fveq2 6756	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝐴‘𝑘) = (𝐴‘𝑁))
42		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝑥↑𝑘) = (𝑥↑𝑁))
43	41, 42	oveq12d 7273	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((𝐴‘𝑘) · (𝑥↑𝑘)) = ((𝐴‘𝑁) · (𝑥↑𝑁)))
44	38, 40, 43	fsumm1 15391	. . . . . . . 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 2778	. . . . . . 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 11317	. . . . . 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 6760	. . . . 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 15076	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
49	25	abscld 15076	. . . . . . 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 15374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
51	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ⁺)
52	51	rpred 12701	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
53	21	abscld 15076	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
54	53, 29	reexpcld 13809	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
55	52, 54	remulcld 10936	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
56	18, 25	fsumabs 15441	. . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
58	27	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
59		nnm1nn0 12204	. . . . . . . . . 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 13809	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
62	57, 61	remulcld 10936	. . . . . . 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 711	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
64	61	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
65	63, 64	remulcld 10936	. . . . . . . . 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 15094	. . . . . . . . . . 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 592	. . . . . . . . . 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 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥↑𝑘) ∈ ℂ)
69	68	abscld 15076	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ∈ ℝ)
70	7, 20	sylan2 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
71	70	absge0d 15084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴‘𝑘)))
72		absexp 14944	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
73	21, 7, 72	syl2an 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
74	53	adantr 480	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
77		max1 12848	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
78	15, 14, 77	sylancr 586	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
79	78	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
80		simprr 769	. . . . . . . . . . . . . . . 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 11063	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
82	75, 53, 81	ltled 11053	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
83	82	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
84		elfzuz3 13182	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
85	84	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
86	74, 83, 85	leexp2ad 13899	. . . . . . . . . . . 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 5092	. . . . . . . . . . 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 11845	. . . . . . . . . 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 5092	. . . . . . . . 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 15439	. . . . . . . 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 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
92	63	recnd 10934	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℂ)
93	18, 91, 92	fsummulc1 15425	. . . . . . . 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 5098	. . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
96		max2 12850	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
97	15, 14, 96	sylancr 586	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
98	97	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
99	95, 76, 53, 98, 80	lelttrd 11063	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
100	1, 99	eqbrtrrid 5106	. . . . . . . . . 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 12752	. . . . . . . . . 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 10936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
104	60	nn0zd 12353	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
105		0red 10909	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
106		0lt1 11427	. . . . . . . . . . . . 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 11066	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
109		expgt0 13744	. . . . . . . . . . 11 ⊢ (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
110	53, 104, 108, 109	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
111		ltmul1 11755	. . . . . . . . . 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 1372	. . . . . . . . 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 10934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
115		expm1t 13739	. . . . . . . . . . . 12 ⊢ (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
116	114, 58, 115	syl2anc 583	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
117	91, 114	mulcomd 10927	. . . . . . . . . . 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 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
119	118	oveq2d 7271	. . . . . . . . 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 10934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
121	120, 114, 91	mulassd 10929	. . . . . . . . 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 2781	. . . . . . . 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 5098	. . . . . . 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 11063	. . . . . 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 11063	. . . . 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 5092	. . . 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 456	. . 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 3107	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
129		breq1 5073	. . 3 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
130	129	rspceaimv 3557	. 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 583	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 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ifcif 4456 class class class wbr 5070 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 ...cfz 13168 ↑cexp 13710 abscabs 14873 Σcsu 15325 Polycply 25250 coeffccoe 25252 degcdgr 25253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-ico 13014 df-fz 13169 df-fzo 13312 df-fl 13440 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126 df-sum 15326 df-0p 24739 df-ply 25254 df-coe 25256 df-dgr 25257

This theorem is referenced by: ftalem2 26128

W3C validator