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Theorem ftalem1 25019
Description: Lemma for fta 25026: "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 12986 . . . . . 6 (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3 ftalem.3 . . . . . . . . 9 (𝜑𝐹 ∈ (Poly‘𝑆))
4 ftalem.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
54coef3 24207 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
63, 5syl 17 . . . . . . . 8 (𝜑𝐴:ℕ0⟶ℂ)
7 elfznn0 12646 . . . . . . . 8 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
8 ffvelrn 6521 . . . . . . . 8 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
96, 7, 8syl2an 495 . . . . . . 7 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
109abscld 14394 . . . . . 6 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
112, 10fsumrecl 14684 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
12 ftalem1.5 . . . . 5 (𝜑𝐸 ∈ ℝ+)
1311, 12rerpdivcld 12116 . . . 4 (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) ∈ ℝ)
141, 13syl5eqel 2843 . . 3 (𝜑𝑇 ∈ ℝ)
15 1re 10251 . . 3 1 ∈ ℝ
16 ifcl 4274 . . 3 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
1714, 15, 16sylancl 697 . 2 (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
183adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
19 simprl 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
20 ftalem.2 . . . . . . . . . . 11 𝑁 = (deg‘𝐹)
214, 20coeid2 24214 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
2218, 19, 21syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
23 ftalem.4 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
2423nnnn0d 11563 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
2524adantr 472 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ0)
26 nn0uz 11935 . . . . . . . . . . 11 0 = (ℤ‘0)
2725, 26syl6eleq 2849 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ‘0))
28 elfznn0 12646 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
296adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ0⟶ℂ)
3029, 8sylan 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
31 expcl 13092 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3219, 31sylan 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3330, 32mulcld 10272 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
3428, 33sylan2 492 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
35 fveq2 6353 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
36 oveq2 6822 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝑥𝑘) = (𝑥𝑁))
3735, 36oveq12d 6832 . . . . . . . . . 10 (𝑘 = 𝑁 → ((𝐴𝑘) · (𝑥𝑘)) = ((𝐴𝑁) · (𝑥𝑁)))
3827, 34, 37fsumm1 14699 . . . . . . . . 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 2794 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4039oveq1d 6829 . . . . . . 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 12986 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
427, 33sylan2 492 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4341, 42fsumcl 14683 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4429, 25ffvelrnd 6524 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴𝑁) ∈ ℂ)
4519, 25expcld 13222 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥𝑁) ∈ ℂ)
4644, 45mulcld 10272 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴𝑁) · (𝑥𝑁)) ∈ ℂ)
4743, 46pncand 10605 . . . . . . 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 2794 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4948fveq2d 6357 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))))
5043abscld 14394 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5142abscld 14394 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5241, 51fsumrecl 14684 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5312adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ+)
5453rpred 12085 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
5519abscld 14394 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
5655, 25reexpcld 13239 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
5754, 56remulcld 10282 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
5841, 42fsumabs 14752 . . . . . 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 472 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
6023adantr 472 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
61 nnm1nn0 11546 . . . . . . . . . 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 13239 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6459, 63remulcld 10282 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6510adantlr 753 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
6663adantr 472 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6765, 66remulcld 10282 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6830, 32absmuld 14412 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
697, 68sylan2 492 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
707, 32sylan2 492 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥𝑘) ∈ ℂ)
7170abscld 14394 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ∈ ℝ)
727, 30sylan2 492 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
7372absge0d 14402 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴𝑘)))
74 absexp 14263 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7519, 7, 74syl2an 495 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7655adantr 472 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
79 max1 12229 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8015, 14, 79sylancr 698 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8180adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
82 simprr 813 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))
8377, 78, 55, 81, 82lelttrd 10407 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
8477, 55, 83ltled 10397 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
8584adantr 472 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
86 elfzuz3 12552 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑘))
8786adantl 473 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑘))
8876, 85, 87leexp2ad 13255 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8975, 88eqbrtrd 4826 . . . . . . . . . . 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 11176 . . . . . . . . . 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 4826 . . . . . . . . 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 14750 . . . . . . . 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 10280 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
9465recnd 10280 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℂ)
9541, 93, 94fsummulc1 14736 . . . . . . . 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 4832 . . . . . . 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 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
98 max2 12231 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9915, 14, 98sylancr 698 . . . . . . . . . . . . 13 (𝜑𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10099adantr 472 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10197, 78, 55, 100, 82lelttrd 10407 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
1021, 101syl5eqbrr 4840 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥))
10359, 55, 53ltdivmuld 12136 . . . . . . . . . 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 222 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)))
10554, 55remulcld 10282 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
10662nn0zd 11692 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
107 0red 10253 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
108 0lt1 10762 . . . . . . . . . . . . 13 0 < 1
109108a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
110107, 77, 55, 109, 83lttrd 10410 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
111 expgt0 13107 . . . . . . . . . . 11 (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
11255, 106, 110, 111syl3anc 1477 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
113 ltmul1 11085 . . . . . . . . . 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 1481 . . . . . . . . 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 222 . . . . . . . 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 10280 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
117 expm1t 13102 . . . . . . . . . . . 12 (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
118116, 60, 117syl2anc 696 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
11993, 116mulcomd 10273 . . . . . . . . . . 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 2794 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
121120oveq2d 6830 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
12254recnd 10280 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
123122, 116, 93mulassd 10275 . . . . . . . . 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 2797 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
125115, 124breqtrrd 4832 . . . . . . 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 10407 . . . . . 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 10407 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12849, 127eqbrtrd 4826 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
129128expr 644 . . 3 ((𝜑𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
130129ralrimiva 3104 . 2 (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
131 breq1 4807 . . . . 5 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
132131imbi1d 330 . . . 4 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → ((𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))) ↔ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))))
133132ralbidv 3124 . . 3 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))) ↔ ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))))
134133rspcev 3449 . 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))))
13517, 130, 134syl2anc 696 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 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  ifcif 4230   class class class wbr 4804  wf 6045  cfv 6049  (class class class)co 6814  cc 10146  cr 10147  0cc0 10148  1c1 10149   + caddc 10151   · cmul 10153   < clt 10286  cle 10287  cmin 10478   / cdiv 10896  cn 11232  0cn0 11504  cz 11589  cuz 11899  +crp 12045  ...cfz 12539  cexp 13074  abscabs 14193  Σcsu 14635  Polycply 24159  coeffccoe 24161  degcdgr 24162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226  ax-addf 10227
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-rp 12046  df-ico 12394  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-rlim 14439  df-sum 14636  df-0p 23656  df-ply 24163  df-coe 24165  df-dgr 24166
This theorem is referenced by:  ftalem2  25020
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