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Theorem ftalem1 25658

Description: Lemma for fta 25665: "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 13336	. . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
4		ftalem.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
5	4	coef3 24829	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7		elfznn0 12995	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
8		ffvelrn 6826	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
9	6, 7, 8	syl2an 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
10	9	abscld 14788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
11	2, 10	fsumrecl 15083	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
12		ftalem1.5	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13	11, 12	rerpdivcld 12450	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) ∈ ℝ)
14	1, 13	eqeltrid 2894	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ)
15		1re 10630	. . 3 ⊢ 1 ∈ ℝ
16		ifcl 4469	. . 3 ⊢ ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
17	14, 15, 16	sylancl 589	. 2 ⊢ (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18		fzfid 13336	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
19	6	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ₀⟶ℂ)
20	19, 8	sylan 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
21		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
22		expcl 13443	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
23	21, 22	sylan 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
24	20, 23	mulcld 10650	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
25	7, 24	sylan2 595	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
26	18, 25	fsumcl 15082	. . . . . . 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 11943	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
29	28	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
30	19, 29	ffvelrnd 6829	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
31	21, 29	expcld 13506	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
32	30, 31	mulcld 10650	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
33	3	adantr 484	. . . . . . . . 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 24836	. . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
36	33, 21, 35	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
37		nn0uz 12268	. . . . . . . . . 10 ⊢ ℕ₀ = (ℤ_≥‘0)
38	29, 37	eleqtrdi 2900	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘0))
39		elfznn0 12995	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
40	39, 24	sylan2 595	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
41		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝐴‘𝑘) = (𝐴‘𝑁))
42		oveq2 7143	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → (𝑥↑𝑘) = (𝑥↑𝑁))
43	41, 42	oveq12d 7153	. . . . . . . . 9 ⊢ (𝑘 = 𝑁 → ((𝐴‘𝑘) · (𝑥↑𝑘)) = ((𝐴‘𝑁) · (𝑥↑𝑁)))
44	38, 40, 43	fsumm1 15098	. . . . . . . 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 2833	. . . . . . 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 11041	. . . . . 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 6649	. . . . 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 14788	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
49	25	abscld 14788	. . . . . . 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 15083	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
51	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ⁺)
52	51	rpred 12419	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
53	21	abscld 14788	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
54	53, 29	reexpcld 13523	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
55	52, 54	remulcld 10660	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
56	18, 25	fsumabs 15148	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
58	27	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
59		nnm1nn0 11926	. . . . . . . . . 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 13523	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
62	57, 61	remulcld 10660	. . . . . . 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 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
64	61	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
65	63, 64	remulcld 10660	. . . . . . . . 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 14806	. . . . . . . . . . 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 595	. . . . . . . . . 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 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥↑𝑘) ∈ ℂ)
69	68	abscld 14788	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ∈ ℝ)
70	7, 20	sylan2 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
71	70	absge0d 14796	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴‘𝑘)))
72		absexp 14656	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
73	21, 7, 72	syl2an 598	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
74	53	adantr 484	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
77		max1 12566	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
78	15, 14, 77	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
79	78	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
80		simprr 772	. . . . . . . . . . . . . . . 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 10787	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
82	75, 53, 81	ltled 10777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
83	82	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
84		elfzuz3 12899	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
85	84	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
86	74, 83, 85	leexp2ad 13613	. . . . . . . . . . . 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 5052	. . . . . . . . . . 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 11569	. . . . . . . . . 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 5052	. . . . . . . . 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 15146	. . . . . . . 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 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
92	63	recnd 10658	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℂ)
93	18, 91, 92	fsummulc1 15132	. . . . . . . 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 5058	. . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
96		max2 12568	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
97	15, 14, 96	sylancr 590	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
98	97	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
99	95, 76, 53, 98, 80	lelttrd 10787	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
100	1, 99	eqbrtrrid 5066	. . . . . . . . . 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 12470	. . . . . . . . . 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 235	. . . . . . . . 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 10660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
104	60	nn0zd 12073	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
105		0red 10633	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
106		0lt1 11151	. . . . . . . . . . . . 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 10790	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
109		expgt0 13458	. . . . . . . . . . 11 ⊢ (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
110	53, 104, 108, 109	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
111		ltmul1 11479	. . . . . . . . . 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 1371	. . . . . . . . 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 235	. . . . . . . 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 10658	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
115		expm1t 13453	. . . . . . . . . . . 12 ⊢ (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
116	114, 58, 115	syl2anc 587	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
117	91, 114	mulcomd 10651	. . . . . . . . . . 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 2833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
119	118	oveq2d 7151	. . . . . . . . 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 10658	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
121	120, 114, 91	mulassd 10653	. . . . . . . . 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 2836	. . . . . . . 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 5058	. . . . . . 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 10787	. . . . . 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 10787	. . . . 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 5052	. . . 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 460	. . 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 3149	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
129		breq1 5033	. . 3 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
130	129	rspceaimv 3576	. 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 587	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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ifcif 4425 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 ↑cexp 13425 abscabs 14585 Σcsu 15034 Polycply 24781 coeffccoe 24783 degcdgr 24784

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788

This theorem is referenced by: ftalem2 25659

W3C validator