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Theorem ftalem1 24799

Description: Lemma for fta 24806: "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	⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
ftalem.4	⊢ (𝜑 → 𝑁 ∈ ℕ)
ftalem1.5	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
ftalem1.6	⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸)

**Assertion**
Ref	Expression
ftalem1	⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (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**
Step	Hyp	Ref	Expression
1		ftalem1.6	. . . 4 ⊢ 𝑇 = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸)
2		fzfid 12772	. . . . . 6 ⊢ (𝜑 → (0...(𝑁 − 1)) ∈ Fin)
3		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
4		ftalem.1	. . . . . . . . . 10 ⊢ 𝐴 = (coeff‘𝐹)
5	4	coef3 23988	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7		elfznn0 12433	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → 𝑘 ∈ ℕ₀)
8		ffvelrn 6357	. . . . . . . 8 ⊢ ((𝐴:ℕ₀⟶ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
9	6, 7, 8	syl2an 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
10	9	abscld 14175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
11	2, 10	fsumrecl 14465	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
12		ftalem1.5	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
13	11, 12	rerpdivcld 11903	. . . 4 ⊢ (𝜑 → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) ∈ ℝ)
14	1, 13	syl5eqel 2705	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℝ)
15		1re 10039	. . 3 ⊢ 1 ∈ ℝ
16		ifcl 4130	. . 3 ⊢ ((𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
17	14, 15, 16	sylancl 694	. 2 ⊢ (𝜑 → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
18	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐹 ∈ (Poly‘𝑆))
19		simprl 794	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑥 ∈ ℂ)
20		ftalem.2	. . . . . . . . . . 11 ⊢ 𝑁 = (deg‘𝐹)
21	4, 20	coeid2 23995	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
22	18, 19, 21	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)))
23		ftalem.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
24	23	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
25	24	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
26		nn0uz 11722	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
27	25, 26	syl6eleq 2711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘0))
28		elfznn0 12433	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
29	6	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐴:ℕ₀⟶ℂ)
30	29, 8	sylan 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝐴‘𝑘) ∈ ℂ)
31		expcl 12878	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
32	19, 31	sylan 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (𝑥↑𝑘) ∈ ℂ)
33	30, 32	mulcld 10060	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
34	28, 33	sylan2 491	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
35		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (𝐴‘𝑘) = (𝐴‘𝑁))
36		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑁 → (𝑥↑𝑘) = (𝑥↑𝑁))
37	35, 36	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑘 = 𝑁 → ((𝐴‘𝑘) · (𝑥↑𝑘)) = ((𝐴‘𝑁) · (𝑥↑𝑁)))
38	27, 34, 37	fsumm1 14480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...𝑁)((𝐴‘𝑘) · (𝑥↑𝑘)) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))))
39	22, 38	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐹‘𝑥) = (Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))))
40	39	oveq1d 6665	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) = ((Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))) − ((𝐴‘𝑁) · (𝑥↑𝑁))))
41		fzfid 12772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (0...(𝑁 − 1)) ∈ Fin)
42	7, 33	sylan2 491	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
43	41, 42	fsumcl 14464	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) ∈ ℂ)
44	29, 25	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
45	19, 25	expcld 13008	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
46	44, 45	mulcld 10060	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
47	43, 46	pncand 10393	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)) + ((𝐴‘𝑁) · (𝑥↑𝑁))) − ((𝐴‘𝑁) · (𝑥↑𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)))
48	40, 47	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) = Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘)))
49	48	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) = (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))))
50	43	abscld 14175	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
51	42	abscld 14175	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
52	41, 51	fsumrecl 14465	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ∈ ℝ)
53	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ⁺)
54	53	rpred 11872	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℝ)
55	19	abscld 14175	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
56	55, 25	reexpcld 13025	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
57	54, 56	remulcld 10070	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
58	41, 42	fsumabs 14533	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))))
59	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ)
60	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑁 ∈ ℕ)
61		nnm1nn0 11334	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
62	60, 61	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℕ₀)
63	55, 62	reexpcld 13025	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
64	59, 63	remulcld 10070	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
65	10	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℝ)
66	63	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ)
67	65, 66	remulcld 10070	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) ∈ ℝ)
68	30, 32	absmuld 14193	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ ℕ₀) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) = ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))))
69	7, 68	sylan2 491	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) = ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))))
70	7, 32	sylan2 491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑥↑𝑘) ∈ ℂ)
71	70	abscld 14175	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ∈ ℝ)
72	7, 30	sylan2 491	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝐴‘𝑘) ∈ ℂ)
73	72	absge0d 14183	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 0 ≤ (abs‘(𝐴‘𝑘)))
74		absexp 14044	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
75	19, 7, 74	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) = ((abs‘𝑥)↑𝑘))
76	55	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘𝑥) ∈ ℝ)
77	15	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ∈ ℝ)
78	17	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ)
79		max1 12016	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
80	15, 14, 79	sylancr 695	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
81	80	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ if(1 ≤ 𝑇, 𝑇, 1))
82		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))
83	77, 78, 55, 81, 82	lelttrd 10195	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 < (abs‘𝑥))
84	77, 55, 83	ltled 10185	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
85	84	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → 1 ≤ (abs‘𝑥))
86		elfzuz3 12339	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
87	86	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ_≥‘𝑘))
88	76, 85, 87	leexp2ad 13041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘𝑥)↑𝑘) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
89	75, 88	eqbrtrd 4675	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝑥↑𝑘)) ≤ ((abs‘𝑥)↑(𝑁 − 1)))
90	71, 66, 65, 73, 89	lemul2ad 10964	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → ((abs‘(𝐴‘𝑘)) · (abs‘(𝑥↑𝑘))) ≤ ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
91	69, 90	eqbrtrd 4675	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ ((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
92	41, 51, 67, 91	fsumle 14531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
93	63	recnd 10068	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
94	65	recnd 10068	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) ∧ 𝑘 ∈ (0...(𝑁 − 1))) → (abs‘(𝐴‘𝑘)) ∈ ℂ)
95	41, 93, 94	fsummulc1 14517	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
96	92, 95	breqtrrd 4681	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) ≤ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))))
97	14	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ∈ ℝ)
98		max2 12018	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
99	15, 14, 98	sylancr 695	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
100	99	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 ≤ if(1 ≤ 𝑇, 𝑇, 1))
101	97, 78, 55, 100, 82	lelttrd 10195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
102	1, 101	syl5eqbrr 4689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) < (abs‘𝑥))
103	59, 55, 53	ltdivmuld 11923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / 𝐸) < (abs‘𝑥) ↔ Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥))))
104	102, 103	mpbid 222	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)))
105	54, 55	remulcld 10070	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · (abs‘𝑥)) ∈ ℝ)
106	62	nn0zd 11480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝑁 − 1) ∈ ℤ)
107		0red 10041	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 ∈ ℝ)
108		0lt1 10550	. . . . . . . . . . . . 13 ⊢ 0 < 1
109	108	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < 1)
110	107, 77, 55, 109, 83	lttrd 10198	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < (abs‘𝑥))
111		expgt0 12893	. . . . . . . . . . 11 ⊢ (((abs‘𝑥) ∈ ℝ ∧ (𝑁 − 1) ∈ ℤ ∧ 0 < (abs‘𝑥)) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
112	55, 106, 110, 111	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 0 < ((abs‘𝑥)↑(𝑁 − 1)))
113		ltmul1 10873	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) ∈ ℝ ∧ (𝐸 · (abs‘𝑥)) ∈ ℝ ∧ (((abs‘𝑥)↑(𝑁 − 1)) ∈ ℝ ∧ 0 < ((abs‘𝑥)↑(𝑁 − 1)))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
114	59, 105, 63, 112, 113	syl112anc 1330	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) < (𝐸 · (abs‘𝑥)) ↔ (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1)))))
115	104, 114	mpbid 222	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
116	55	recnd 10068	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
117		expm1t 12888	. . . . . . . . . . . 12 ⊢ (((abs‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
118	116, 60, 117	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)))
119	93, 116	mulcomd 10061	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (((abs‘𝑥)↑(𝑁 − 1)) · (abs‘𝑥)) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
120	118, 119	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) = ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1))))
121	120	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = (𝐸 · ((abs‘𝑥) · ((abs‘𝑥)↑(𝑁 − 1)))))
122	54	recnd 10068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → 𝐸 ∈ ℂ)
123	122, 116, 93	mulassd 10063	. . . . . . . . 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)))))
124	121, 123	eqtr4d 2659	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (𝐸 · ((abs‘𝑥)↑𝑁)) = ((𝐸 · (abs‘𝑥)) · ((abs‘𝑥)↑(𝑁 − 1))))
125	115, 124	breqtrrd 4681	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) · ((abs‘𝑥)↑(𝑁 − 1))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
126	52, 64, 57, 96, 125	lelttrd 10195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘((𝐴‘𝑘) · (𝑥↑𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
127	50, 52, 57, 58, 126	lelttrd 10195	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘Σ𝑘 ∈ (0...(𝑁 − 1))((𝐴‘𝑘) · (𝑥↑𝑘))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
128	49, 127	eqbrtrd 4675	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ ℂ ∧ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))
129	128	expr 643	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
130	129	ralrimiva 2966	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
131		breq1 4656	. . . . 5 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (𝑟 < (abs‘𝑥) ↔ if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥)))
132	131	imbi1d 331	. . . 4 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → ((𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))) ↔ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))))
133	132	ralbidv 2986	. . 3 ⊢ (𝑟 = if(1 ≤ 𝑇, 𝑇, 1) → (∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))) ↔ ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))))
134	133	rspcev 3309	. 2 ⊢ ((if(1 ≤ 𝑇, 𝑇, 1) ∈ ℝ ∧ ∀𝑥 ∈ ℂ (if(1 ≤ 𝑇, 𝑇, 1) < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁)))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))
135	17, 130, 134	syl2anc 693	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (𝐸 · ((abs‘𝑥)↑𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ifcif 4086 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ℤ_≥cuz 11687 ℝ⁺crp 11832 ...cfz 12326 ↑cexp 12860 abscabs 13974 Σcsu 14416 Polycply 23940 coeffccoe 23942 degcdgr 23943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-0p 23437 df-ply 23944 df-coe 23946 df-dgr 23947

This theorem is referenced by: ftalem2 24800

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