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Theorem ftalem1 24486
Description: Lemma for fta 24495: "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 12502 . . . . . 6 (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3 ftalem.3 . . . . . . . . 9 (𝜑𝐹 ∈ (Poly‘𝑆))
4 ftalem.1 . . . . . . . . . 10 𝐴 = (coeff‘𝐹)
54coef3 23676 . . . . . . . . 9 (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ)
63, 5syl 17 . . . . . . . 8 (𝜑𝐴:ℕ0⟶ℂ)
7 elfznn0 12170 . . . . . . . 8 (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ0)
8 ffvelrn 6149 . . . . . . . 8 ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
96, 7, 8syl2an 492 . . . . . . 7 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
109abscld 13882 . . . . . 6 ((𝜑𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
112, 10fsumrecl 14181 . . . . 5 (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
12 ftalem1.5 . . . . 5 (𝜑𝐸 ∈ ℝ+)
1311, 12rerpdivcld 11645 . . . 4 (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) ∈ ℝ)
141, 13syl5eqel 2596 . . 3 (𝜑𝑇 ∈ ℝ)
15 1re 9794 . . 3 1 ∈ ℝ
16 ifcl 3983 . . 3 ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
1714, 15, 16sylancl 692 . 2 (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
183adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
19 simprl 789 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
20 ftalem.2 . . . . . . . . . . 11 𝑁 = (deg‘𝐹)
214, 20coeid2 23683 . . . . . . . . . 10 ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
2218, 19, 21syl2anc 690 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴𝑘) · (𝑥𝑘)))
23 ftalem.4 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℕ)
2423nnnn0d 11106 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
2524adantr 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ0)
26 nn0uz 11462 . . . . . . . . . . 11 0 = (ℤ‘0)
2725, 26syl6eleq 2602 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ‘0))
28 elfznn0 12170 . . . . . . . . . . 11 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
296adantr 479 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ0⟶ℂ)
3029, 8sylan 486 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝐴𝑘) ∈ ℂ)
31 expcl 12608 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3219, 31sylan 486 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (𝑥𝑘) ∈ ℂ)
3330, 32mulcld 9815 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
3428, 33sylan2 489 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
35 fveq2 5987 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝐴𝑘) = (𝐴𝑁))
36 oveq2 6434 . . . . . . . . . . 11 (𝑘 = 𝑁 → (𝑥𝑘) = (𝑥𝑁))
3735, 36oveq12d 6444 . . . . . . . . . 10 (𝑘 = 𝑁 → ((𝐴𝑘) · (𝑥𝑘)) = ((𝐴𝑁) · (𝑥𝑁)))
3827, 34, 37fsumm1 14193 . . . . . . . . 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 2548 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) + ((𝐴𝑁) · (𝑥𝑁))))
4039oveq1d 6441 . . . . . . 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 12502 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
427, 33sylan2 489 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4341, 42fsumcl 14180 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)) ∈ ℂ)
4429, 25ffvelrnd 6152 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴𝑁) ∈ ℂ)
4519, 25expcld 12738 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥𝑁) ∈ ℂ)
4644, 45mulcld 9815 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴𝑁) · (𝑥𝑁)) ∈ ℂ)
4743, 46pncand 10144 . . . . . . 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 2548 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘)))
4948fveq2d 5991 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))))
5043abscld 13882 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5142abscld 13882 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5241, 51fsumrecl 14181 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴𝑘) · (𝑥𝑘))) ∈ ℝ)
5312adantr 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ+)
5453rpred 11614 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
5519abscld 13882 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
5655, 25reexpcld 12755 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
5754, 56remulcld 9825 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
5841, 42fsumabs 14243 . . . . . 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 479 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) ∈ ℝ)
6023adantr 479 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
61 nnm1nn0 11089 . . . . . . . . . 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 12755 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6459, 63remulcld 9825 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6510adantlr 746 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℝ)
6663adantr 479 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
6765, 66remulcld 9825 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
6830, 32absmuld 13900 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ0) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
697, 68sylan2 489 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴𝑘) · (𝑥𝑘))) = ((abs‘(𝐴𝑘)) · (abs‘(𝑥𝑘))))
707, 32sylan2 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥𝑘) ∈ ℂ)
7170abscld 13882 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) ∈ ℝ)
727, 30sylan2 489 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴𝑘) ∈ ℂ)
7372absge0d 13890 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴𝑘)))
74 absexp 13751 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7519, 7, 74syl2an 492 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥𝑘)) = ((abs‘𝑥)↑𝑘))
7655adantr 479 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
79 max1 11759 . . . . . . . . . . . . . . . . . 18 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8015, 14, 79sylancr 693 . . . . . . . . . . . . . . . . 17 (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
8180adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
82 simprr 791 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))
8377, 78, 55, 81, 82lelttrd 9946 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
8477, 55, 83ltled 9936 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
8584adantr 479 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
86 elfzuz3 12078 . . . . . . . . . . . . . 14 (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑘))
8786adantl 480 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑘))
8876, 85, 87leexp2ad 12771 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
8975, 88eqbrtrd 4503 . . . . . . . . . . 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 10714 . . . . . . . . . 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 4503 . . . . . . . . 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 14241 . . . . . . . 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 9823 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
9465recnd 9823 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴𝑘)) ∈ ℂ)
9541, 93, 94fsummulc1 14228 . . . . . . . 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 4509 . . . . . . 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 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
98 max2 11761 . . . . . . . . . . . . . 14 ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
9915, 14, 98sylancr 693 . . . . . . . . . . . . 13 (𝜑𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10099adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
10197, 78, 55, 100, 82lelttrd 9946 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
1021, 101syl5eqbrr 4517 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) / 𝐸) < (abs‘𝑥))
10359, 55, 53ltdivmuld 11665 . . . . . . . . . 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 220 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴𝑘)) < (𝐸 · (abs‘𝑥)))
10554, 55remulcld 9825 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
10662nn0zd 11220 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
107 0red 9796 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
108 0lt1 10299 . . . . . . . . . . . . 13 0 < 1
109108a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
110107, 77, 55, 109, 83lttrd 9949 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
111 expgt0 12623 . . . . . . . . . . 11 (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
11255, 106, 110, 111syl3anc 1317 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
113 ltmul1 10622 . . . . . . . . . 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 1321 . . . . . . . . 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 220 . . . . . . . 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 9823 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
117 expm1t 12618 . . . . . . . . . . . 12 (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
118116, 60, 117syl2anc 690 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
11993, 116mulcomd 9816 . . . . . . . . . . 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 2548 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
121120oveq2d 6442 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
12254recnd 9823 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
123122, 116, 93mulassd 9818 . . . . . . . . 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 2551 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
125115, 124breqtrrd 4509 . . . . . . 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 9946 . . . . . 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 9946 . . . . 5 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴𝑘) · (𝑥𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
12849, 127eqbrtrd 4503 . . . 4 ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
129128expr 640 . . 3 ((𝜑𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
130129ralrimiva 2853 . 2 (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹𝑥) − ((𝐴𝑁) · (𝑥𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
131 breq1 4484 . . . . 5 (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
132131imbi1d 329 . . . 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 2873 . . 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 3186 . 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 690 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 194  wa 382   = wceq 1474  wcel 1938  wral 2800  wrex 2801  ifcif 3939   class class class wbr 4481  wf 5685  cfv 5689  (class class class)co 6426  cc 9689  cr 9690  0cc0 9691  1c1 9692   + caddc 9694   · cmul 9696   < clt 9829  cle 9830  cmin 10017   / cdiv 10433  cn 10775  0cn0 11047  cz 11118  cuz 11427  +crp 11574  ...cfz 12065  cexp 12590  abscabs 13681  Σcsu 14133  Polycply 23628  coeffccoe 23630  degcdgr 23631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-inf2 8297  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768  ax-pre-sup 9769  ax-addf 9770
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-of 6671  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-oadd 7327  df-er 7505  df-map 7622  df-pm 7623  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-sup 8107  df-inf 8108  df-oi 8174  df-card 8524  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-div 10434  df-nn 10776  df-2 10834  df-3 10835  df-n0 11048  df-z 11119  df-uz 11428  df-rp 11575  df-ico 11921  df-fz 12066  df-fzo 12203  df-fl 12323  df-seq 12532  df-exp 12591  df-hash 12848  df-cj 13546  df-re 13547  df-im 13548  df-sqrt 13682  df-abs 13683  df-clim 13933  df-rlim 13934  df-sum 14134  df-0p 23118  df-ply 23632  df-coe 23634  df-dgr 23635
This theorem is referenced by:  ftalem2  24487
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