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Theorem ftalem1 25665
 Description: Lemma for fta 25672: "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 13345 . . . . . 6 (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3 ftalem.3 . . . . . . . . 9 (𝜑𝐹 ∈ (Poly‘𝑆))
4 ftalem.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
54coef3 24836 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
63, 5syl 17 . . . . . . . 8 (𝜑𝐴:ℕ0⟶ℂ)
7 elfznn0 13004 . . . . . . . 8 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
8 ffvelrn 6840 . . . . . . . 8 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
96, 7, 8syl2an 598 . . . . . . 7 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
109abscld 14796 . . . . . 6 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
112, 10fsumrecl 15091 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
12 ftalem1.5 . . . . 5 (𝜑𝐸 ∈ ℝ+)
1311, 12rerpdivcld 12459 . . . 4 (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) ∈ ℝ)
141, 13eqeltrid 2920 . . 3 (𝜑𝑇 ∈ ℝ)
15 1re 10639 . . 3 1 ∈ ℝ
16 ifcl 4494 . . 3 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
1714, 15, 16sylancl 589 . 2 (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18 fzfid 13345 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
196adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ0⟶ℂ)
2019, 8sylan 583 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
21 simprl 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
22 expcl 13452 . . . . . . . . . . 11 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
2321, 22sylan 583 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
2420, 23mulcld 10659 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
257, 24sylan2 595 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
2618, 25fsumcl 15090 . . . . . . 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 11952 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ0)
2928adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ0)
3019, 29ffvelrnd 6843 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴𝑁) ∈ ℂ)
3121, 29expcld 13515 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥𝑁) ∈ ℂ)
3230, 31mulcld 10659 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴𝑁) · (𝑥𝑁)) ∈ ℂ)
333adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
34 ftalem.2 . . . . . . . . . 10 𝑁 = (deg‘𝐹)
354, 34coeid2 24843 . . . . . . . . 9 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
3633, 21, 35syl2anc 587 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
37 nn0uz 12277 . . . . . . . . . 10 0 = (ℤ‘0)
3829, 37eleqtrdi 2926 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ‘0))
39 elfznn0 13004 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
4039, 24sylan2 595 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
41 fveq2 6661 . . . . . . . . . 10 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
42 oveq2 7157 . . . . . . . . . 10 (𝑘 = 𝑁 → (𝑥𝑘) = (𝑥𝑁))
4341, 42oveq12d 7167 . . . . . . . . 9 (𝑘 = 𝑁 → ((𝐴𝑘) · (𝑥𝑘)) = ((𝐴𝑁) · (𝑥𝑁)))
4438, 40, 43fsumm1 15106 . . . . . . . 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 2859 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4626, 32, 45mvrraddd 11050 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4746fveq2d 6665 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))))
4826abscld 14796 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
4925abscld 14796 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5018, 49fsumrecl 15091 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5112adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ+)
5251rpred 12428 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
5321abscld 14796 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
5453, 29reexpcld 13532 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
5552, 54remulcld 10669 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
5618, 25fsumabs 15156 . . . . . 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 484 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
5827adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
59 nnm1nn0 11935 . . . . . . . . . 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 13532 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6257, 61remulcld 10669 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6310adantlr 714 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
6461adantr 484 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6563, 64remulcld 10669 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6620, 23absmuld 14814 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
677, 66sylan2 595 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
687, 23sylan2 595 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥𝑘) ∈ ℂ)
6968abscld 14796 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ∈ ℝ)
707, 20sylan2 595 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
7170absge0d 14804 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴𝑘)))
72 absexp 14664 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7321, 7, 72syl2an 598 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7453adantr 484 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
77 max1 12575 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
7815, 14, 77sylancr 590 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
7978adantr 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))
8175, 76, 53, 79, 80lelttrd 10796 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
8275, 53, 81ltled 10786 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
8382adantr 484 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
84 elfzuz3 12908 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑘))
8584adantl 485 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑘))
8674, 83, 85leexp2ad 13622 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8773, 86eqbrtrd 5074 . . . . . . . . . . 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 11578 . . . . . . . . . 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 5074 . . . . . . . . 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 15154 . . . . . . . 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 10667 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
9263recnd 10667 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℂ)
9318, 91, 92fsummulc1 15140 . . . . . . . 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 5080 . . . . . . 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 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
96 max2 12577 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9715, 14, 96sylancr 590 . . . . . . . . . . . . 13 (𝜑𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9897adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9995, 76, 53, 98, 80lelttrd 10796 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
1001, 99eqbrtrrid 5088 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥))
10157, 53, 51ltdivmuld 12479 . . . . . . . . . 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 235 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)))
10352, 53remulcld 10669 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
10460nn0zd 12082 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
105 0red 10642 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
106 0lt1 11160 . . . . . . . . . . . . 13 0 < 1
107106a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
108105, 75, 53, 107, 81lttrd 10799 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
109 expgt0 13467 . . . . . . . . . . 11 (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
11053, 104, 108, 109syl3anc 1368 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
111 ltmul1 11488 . . . . . . . . . 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 1371 . . . . . . . . 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 235 . . . . . . . 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 10667 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
115 expm1t 13462 . . . . . . . . . . . 12 (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
116114, 58, 115syl2anc 587 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
11791, 114mulcomd 10660 . . . . . . . . . . 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 2859 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
119118oveq2d 7165 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
12052recnd 10667 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
121120, 114, 91mulassd 10662 . . . . . . . . 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 2862 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
123113, 122breqtrrd 5080 . . . . . . 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 10796 . . . . . 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 10796 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12647, 125eqbrtrd 5074 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
127126expr 460 . . 3 ((𝜑𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
128127ralrimiva 3177 . 2 (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
129 breq1 5055 . . 3 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
130129rspceaimv 3614 . 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 587 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ∃wrex 3134  ifcif 4450   class class class wbr 5052  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149  ℂcc 10533  ℝcr 10534  0cc0 10535  1c1 10536   + caddc 10538   · cmul 10540   < clt 10673   ≤ cle 10674   − cmin 10868   / cdiv 11295  ℕcn 11634  ℕ0cn0 11894  ℤcz 11978  ℤ≥cuz 12240  ℝ+crp 12386  ...cfz 12894  ↑cexp 13434  abscabs 14593  Σcsu 15042  Polycply 24788  coeffccoe 24790  degcdgr 24791 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-rp 12387  df-ico 12741  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-0p 24281  df-ply 24792  df-coe 24794  df-dgr 24795 This theorem is referenced by:  ftalem2  25666
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