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Theorem ftalem1 26422
Description: Lemma for fta 26429: "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 (𝜑𝐹 ∈ (Poly‘𝑆))
ftalem.4 (𝜑𝑁 ∈ ℕ)
ftalem1.5 (𝜑𝐸 ∈ ℝ+)
ftalem1.6 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸)
Assertion
Ref Expression
ftalem1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (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
StepHypRef Expression
1 ftalem1.6 . . . 4 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸)
2 fzfid 13878 . . . . . 6 (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3 ftalem.3 . . . . . . . . 9 (𝜑𝐹 ∈ (Poly‘𝑆))
4 ftalem.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
54coef3 25593 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
63, 5syl 17 . . . . . . . 8 (𝜑𝐴:ℕ0⟶ℂ)
7 elfznn0 13534 . . . . . . . 8 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
8 ffvelcdm 7032 . . . . . . . 8 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
96, 7, 8syl2an 596 . . . . . . 7 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
109abscld 15321 . . . . . 6 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
112, 10fsumrecl 15619 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
12 ftalem1.5 . . . . 5 (𝜑𝐸 ∈ ℝ+)
1311, 12rerpdivcld 12988 . . . 4 (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) ∈ ℝ)
141, 13eqeltrid 2842 . . 3 (𝜑𝑇 ∈ ℝ)
15 1re 11155 . . 3 1 ∈ ℝ
16 ifcl 4531 . . 3 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
1714, 15, 16sylancl 586 . 2 (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18 fzfid 13878 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
196adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ0⟶ℂ)
2019, 8sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
21 simprl 769 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
22 expcl 13985 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
2321, 22sylan 580 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
2420, 23mulcld 11175 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
257, 24sylan2 593 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
2618, 25fsumcl 15618 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
27 ftalem.4 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ)
2827nnnn0d 12473 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
2928adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ0)
3019, 29ffvelcdmd 7036 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴𝑁) ∈ ℂ)
3121, 29expcld 14051 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥𝑁) ∈ ℂ)
3230, 31mulcld 11175 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴𝑁) · (𝑥𝑁)) ∈ ℂ)
333adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
34 ftalem.2 . . . . . . . . . 10 𝑁 = (deg‘𝐹)
354, 34coeid2 25600 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
3633, 21, 35syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
37 nn0uz 12805 . . . . . . . . . 10 0 = (ℤ‘0)
3829, 37eleqtrdi 2848 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ‘0))
39 elfznn0 13534 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
4039, 24sylan2 593 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
41 fveq2 6842 . . . . . . . . . 10 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
42 oveq2 7365 . . . . . . . . . 10 (𝑘 = 𝑁 → (𝑥𝑘) = (𝑥𝑁))
4341, 42oveq12d 7375 . . . . . . . . 9 (𝑘 = 𝑁 → ((𝐴𝑘) · (𝑥𝑘)) = ((𝐴𝑁) · (𝑥𝑁)))
4438, 40, 43fsumm1 15636 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4536, 44eqtrd 2776 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4626, 32, 45mvrraddd 11567 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4746fveq2d 6846 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))))
4826abscld 15321 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
4925abscld 15321 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5018, 49fsumrecl 15619 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5112adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ+)
5251rpred 12957 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
5321abscld 15321 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
5453, 29reexpcld 14068 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
5552, 54remulcld 11185 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
5618, 25fsumabs 15686 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))))
5711adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
5827adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
59 nnm1nn0 12454 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
6058, 59syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℕ0)
6153, 60reexpcld 14068 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6257, 61remulcld 11185 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6310adantlr 713 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
6461adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6563, 64remulcld 11185 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6620, 23absmuld 15339 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
677, 66sylan2 593 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
687, 23sylan2 593 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥𝑘) ∈ ℂ)
6968abscld 15321 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ∈ ℝ)
707, 20sylan2 593 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
7170absge0d 15329 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴𝑘)))
72 absexp 15189 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7321, 7, 72syl2an 596 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7453adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘𝑥) ∈ ℝ)
7515a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ∈ ℝ)
7617adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
77 max1 13104 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
7815, 14, 77sylancr 587 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
7978adantr 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))
8175, 76, 53, 79, 80lelttrd 11313 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
8275, 53, 81ltled 11303 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
8382adantr 481 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
84 elfzuz3 13438 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑘))
8584adantl 482 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑘))
8674, 83, 85leexp2ad 14157 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8773, 86eqbrtrd 5127 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8869, 64, 63, 71, 87lemul2ad 12095 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))) ≤ ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
8967, 88eqbrtrd 5127 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ≤ ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
9018, 49, 65, 89fsumle 15684 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
9161recnd 11183 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
9263recnd 11183 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℂ)
9318, 91, 92fsummulc1 15670 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
9490, 93breqtrrd 5133 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ≤ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
9514adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
96 max2 13106 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9715, 14, 96sylancr 587 . . . . . . . . . . . . 13 (𝜑𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9897adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9995, 76, 53, 98, 80lelttrd 11313 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
1001, 99eqbrtrrid 5141 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥))
10157, 53, 51ltdivmuld 13008 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥) ↔ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥))))
102100, 101mpbid 231 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)))
10352, 53remulcld 11185 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
10460nn0zd 12525 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
105 0red 11158 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
106 0lt1 11677 . . . . . . . . . . . . 13 0 < 1
107106a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
108105, 75, 53, 107, 81lttrd 11316 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
109 expgt0 14001 . . . . . . . . . . 11 (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
11053, 104, 108, 109syl3anc 1371 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
111 ltmul1 12005 . . . . . . . . . 10 ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ ∧ (𝐸 · (abs‘𝑥)) ∈ ℝ ∧ (((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ ∧ 0 < ((abs‘𝑥)↑(𝑁 − 1)))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
11257, 103, 61, 110, 111syl112anc 1374 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
113102, 112mpbid 231 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
11453recnd 11183 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
115 expm1t 13996 . . . . . . . . . . . 12 (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
116114, 58, 115syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
11791, 114mulcomd 11176 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
118116, 117eqtrd 2776 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
119118oveq2d 7373 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
12052recnd 11183 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
121120, 114, 91mulassd 11178 . . . . . . . . 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)))))
122119, 121eqtr4d 2779 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
123113, 122breqtrrd 5133 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12450, 62, 55, 94, 123lelttrd 11313 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12548, 50, 55, 56, 124lelttrd 11313 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12647, 125eqbrtrd 5127 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
127126expr 457 . . 3 ((𝜑𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
128127ralrimiva 3143 . 2 (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
129 breq1 5108 . . 3 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
130129rspceaimv 3585 . 2 ((if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ ∧ ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
13117, 128, 130syl2anc 584 1 (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ifcif 4486   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  0cn0 12413  cz 12499  cuz 12763  +crp 12915  ...cfz 13424  cexp 13967  abscabs 15119  Σcsu 15570  Polycply 25545  coeffccoe 25547  degcdgr 25548
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-0p 25034  df-ply 25549  df-coe 25551  df-dgr 25552
This theorem is referenced by:  ftalem2  26423
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