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Theorem ftalem1 25251
Description: Lemma for fta 25258: "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 13091 . . . . . 6 (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3 ftalem.3 . . . . . . . . 9 (𝜑𝐹 ∈ (Poly‘𝑆))
4 ftalem.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
54coef3 24425 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
63, 5syl 17 . . . . . . . 8 (𝜑𝐴:ℕ0⟶ℂ)
7 elfznn0 12751 . . . . . . . 8 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
8 ffvelrn 6621 . . . . . . . 8 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
96, 7, 8syl2an 589 . . . . . . 7 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
109abscld 14583 . . . . . 6 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
112, 10fsumrecl 14872 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
12 ftalem1.5 . . . . 5 (𝜑𝐸 ∈ ℝ+)
1311, 12rerpdivcld 12212 . . . 4 (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) ∈ ℝ)
141, 13syl5eqel 2863 . . 3 (𝜑𝑇 ∈ ℝ)
15 1re 10376 . . 3 1 ∈ ℝ
16 ifcl 4351 . . 3 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
1714, 15, 16sylancl 580 . 2 (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
183adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
19 simprl 761 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
20 ftalem.2 . . . . . . . . . . 11 𝑁 = (deg‘𝐹)
214, 20coeid2 24432 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
2218, 19, 21syl2anc 579 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
23 ftalem.4 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
2423nnnn0d 11702 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
2524adantr 474 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ0)
26 nn0uz 12028 . . . . . . . . . . 11 0 = (ℤ‘0)
2725, 26syl6eleq 2869 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ‘0))
28 elfznn0 12751 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
296adantr 474 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ0⟶ℂ)
3029, 8sylan 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
31 expcl 13196 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3219, 31sylan 575 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3330, 32mulcld 10397 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
3428, 33sylan2 586 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
35 fveq2 6446 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
36 oveq2 6930 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝑥𝑘) = (𝑥𝑁))
3735, 36oveq12d 6940 . . . . . . . . . 10 (𝑘 = 𝑁 → ((𝐴𝑘) · (𝑥𝑘)) = ((𝐴𝑁) · (𝑥𝑁)))
3827, 34, 37fsumm1 14887 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
3922, 38eqtrd 2814 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4039oveq1d 6937 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = ((Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))) − ((𝐴𝑁) · (𝑥𝑁))))
41 fzfid 13091 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
427, 33sylan2 586 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4341, 42fsumcl 14871 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4429, 25ffvelrnd 6624 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴𝑁) ∈ ℂ)
4519, 25expcld 13327 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥𝑁) ∈ ℂ)
4644, 45mulcld 10397 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴𝑁) · (𝑥𝑁)) ∈ ℂ)
4743, 46pncand 10735 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4840, 47eqtrd 2814 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4948fveq2d 6450 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))))
5043abscld 14583 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5142abscld 14583 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5241, 51fsumrecl 14872 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5312adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ+)
5453rpred 12181 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
5519abscld 14583 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
5655, 25reexpcld 13344 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
5754, 56remulcld 10407 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
5841, 42fsumabs 14937 . . . . . 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))))
5911adantr 474 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
6023adantr 474 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
61 nnm1nn0 11685 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
6260, 61syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℕ0)
6355, 62reexpcld 13344 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6459, 63remulcld 10407 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6510adantlr 705 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
6663adantr 474 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6765, 66remulcld 10407 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6830, 32absmuld 14601 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
697, 68sylan2 586 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
707, 32sylan2 586 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥𝑘) ∈ ℂ)
7170abscld 14583 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ∈ ℝ)
727, 30sylan2 586 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
7372absge0d 14591 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴𝑘)))
74 absexp 14451 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7519, 7, 74syl2an 589 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7655adantr 474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘𝑥) ∈ ℝ)
7715a1i 11 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ∈ ℝ)
7817adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
79 max1 12328 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8015, 14, 79sylancr 581 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8180adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
82 simprr 763 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))
8377, 78, 55, 81, 82lelttrd 10534 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
8477, 55, 83ltled 10524 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
8584adantr 474 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
86 elfzuz3 12656 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑘))
8786adantl 475 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑘))
8876, 85, 87leexp2ad 13362 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8975, 88eqbrtrd 4908 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
9071, 66, 65, 73, 89lemul2ad 11318 . . . . . . . . . 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))))
9169, 90eqbrtrd 4908 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ≤ ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
9241, 51, 67, 91fsumle 14935 . . . . . . . 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))))
9363recnd 10405 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
9465recnd 10405 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℂ)
9541, 93, 94fsummulc1 14921 . . . . . . . 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))))
9692, 95breqtrrd 4914 . . . . . . 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))))
9714adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
98 max2 12330 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9915, 14, 98sylancr 581 . . . . . . . . . . . . 13 (𝜑𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10099adantr 474 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10197, 78, 55, 100, 82lelttrd 10534 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
1021, 101syl5eqbrr 4922 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥))
10359, 55, 53ltdivmuld 12232 . . . . . . . . . 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))))
104102, 103mpbid 224 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)))
10554, 55remulcld 10407 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
10662nn0zd 11832 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
107 0red 10380 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
108 0lt1 10897 . . . . . . . . . . . . 13 0 < 1
109108a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
110107, 77, 55, 109, 83lttrd 10537 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
111 expgt0 13211 . . . . . . . . . . 11 (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
11255, 106, 110, 111syl3anc 1439 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
113 ltmul1 11227 . . . . . . . . . 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)))))
11459, 105, 63, 112, 113syl112anc 1442 . . . . . . . . 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)))))
115104, 114mpbid 224 . . . . . . . 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))))
11655recnd 10405 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
117 expm1t 13206 . . . . . . . . . . . 12 (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
118116, 60, 117syl2anc 579 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
11993, 116mulcomd 10398 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
120118, 119eqtrd 2814 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
121120oveq2d 6938 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
12254recnd 10405 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
123122, 116, 93mulassd 10400 . . . . . . . . 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)))))
124121, 123eqtr4d 2817 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
125115, 124breqtrrd 4914 . . . . . . 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)))
12652, 64, 57, 96, 125lelttrd 10534 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12750, 52, 57, 58, 126lelttrd 10534 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12849, 127eqbrtrd 4908 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
129128expr 450 . . 3 ((𝜑𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
130129ralrimiva 3148 . 2 (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
131 breq1 4889 . . 3 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
132131rspceaimv 3519 . 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))))
13317, 130, 132syl2anc 579 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 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  ifcif 4307   class class class wbr 4886  wf 6131  cfv 6135  (class class class)co 6922  cc 10270  cr 10271  0cc0 10272  1c1 10273   + caddc 10275   · cmul 10277   < clt 10411  cle 10412  cmin 10606   / cdiv 11032  cn 11374  0cn0 11642  cz 11728  cuz 11992  +crp 12137  ...cfz 12643  cexp 13178  abscabs 14381  Σcsu 14824  Polycply 24377  coeffccoe 24379  degcdgr 24380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350  ax-addf 10351
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-sup 8636  df-inf 8637  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-ico 12493  df-fz 12644  df-fzo 12785  df-fl 12912  df-seq 13120  df-exp 13179  df-hash 13436  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-clim 14627  df-rlim 14628  df-sum 14825  df-0p 23874  df-ply 24381  df-coe 24383  df-dgr 24384
This theorem is referenced by:  ftalem2  25252
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