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Theorem ftalem2 27006

Description: Lemma for fta 27012. There exists some 𝑟 such that 𝐹 has magnitude greater than 𝐹(0) outside the closed ball B(0,r). (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)
ftalem2.5	⊢ 𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))
ftalem2.6	⊢ 𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2))

**Assertion**
Ref	Expression
ftalem2	⊢ (𝜑 → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))

Distinct variable groups: 𝑠,𝑟,𝑥,𝐴 𝑁,𝑟,𝑠,𝑥 𝐹,𝑟,𝑠,𝑥 𝜑,𝑠,𝑥 𝑆,𝑠 𝑇,𝑟,𝑥 𝑈,𝑟,𝑥 Allowed substitution hints: 𝜑(𝑟) 𝑆(𝑥,𝑟) 𝑇(𝑠) 𝑈(𝑠)

Proof of Theorem ftalem2 Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftalem.1	. . 3 ⊢ 𝐴 = (coeff‘𝐹)
2		ftalem.2	. . 3 ⊢ 𝑁 = (deg‘𝐹)
3		ftalem.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
4		ftalem.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	1	coef3 26159	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7	4	nnnn0d 12437	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	6, 7	ffvelcdmd 7013	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ∈ ℂ)
9	4	nnne0d 12170	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
10	2, 1	dgreq0 26193	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
11		fveq2 6817	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
12		dgr0 26190	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
13	11, 12	eqtrdi 2782	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
14	2, 13	eqtrid 2778	. . . . . . . . 9 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
15	10, 14	biimtrrdi 254	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
16	3, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
17	16	necon3d 2949	. . . . . 6 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
18	9, 17	mpd 15	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
19	8, 18	absrpcld 15353	. . . 4 ⊢ (𝜑 → (abs‘(𝐴‘𝑁)) ∈ ℝ⁺)
20	19	rphalfcld 12941	. . 3 ⊢ (𝜑 → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ⁺)
21		2fveq3 6822	. . . . 5 ⊢ (𝑛 = 𝑘 → (abs‘(𝐴‘𝑛)) = (abs‘(𝐴‘𝑘)))
22	21	cbvsumv 15598	. . . 4 ⊢ Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑛)) = Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘))
23	22	oveq1i 7351	. . 3 ⊢ (Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑛)) / ((abs‘(𝐴‘𝑁)) / 2)) = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / ((abs‘(𝐴‘𝑁)) / 2))
24	1, 2, 3, 4, 20, 23	ftalem1 27005	. 2 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))))
25		ftalem2.5	. . . . . 6 ⊢ 𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))
26		ftalem2.6	. . . . . . . . 9 ⊢ 𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2))
27		plyf 26125	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
28	3, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
29		0cn 11099	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
30		ffvelcdm 7009	. . . . . . . . . . . 12 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
31	28, 29, 30	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
32	31	abscld 15341	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ)
33	32, 20	rerpdivcld 12960	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ∈ ℝ)
34	26, 33	eqeltrid 2835	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ)
36		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
37		1re 11107	. . . . . . . 8 ⊢ 1 ∈ ℝ
38		ifcl 4516	. . . . . . . 8 ⊢ ((𝑠 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ)
39	36, 37, 38	sylancl 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ)
40	35, 39	ifcld 4517	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) ∈ ℝ)
41	25, 40	eqeltrid 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ)
42		0red 11110	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
43		1red 11108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
44		0lt1 11634	. . . . . . 7 ⊢ 0 < 1
45	44	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 1)
46		max1 13079	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1))
47	37, 36, 46	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1))
48		max1 13079	. . . . . . . . 9 ⊢ ((if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ ∧ 𝑇 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)))
49	39, 35, 48	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)))
50	49, 25	breqtrrdi 5128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑈)
51	43, 39, 41, 47, 50	letrd 11265	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ 𝑈)
52	42, 43, 41, 45, 51	ltletrd 11268	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 𝑈)
53	41, 52	elrpd 12926	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ⁺)
54		max2 13081	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1))
55	37, 36, 54	sylancr 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1))
56	36, 39, 41, 55, 50	letrd 11265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ 𝑈)
57	56	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ≤ 𝑈)
58		abscl 15180	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
59		lelttr 11198	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (abs‘𝑥) ∈ ℝ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥)))
60	36, 41, 58, 59	syl2an3an 1424	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥)))
61	57, 60	mpand 695	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (𝑈 < (abs‘𝑥) → 𝑠 < (abs‘𝑥)))
62	61	imim1d 82	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))))
63	28	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝐹:ℂ⟶ℂ)
64		simprl 770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑥 ∈ ℂ)
65	63, 64	ffvelcdmd 7013	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐹‘𝑥) ∈ ℂ)
66	8	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
67	7	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
68	64, 67	expcld 14048	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
69	66, 68	mulcld 11127	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
70	65, 69	subcld 11467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
71	70	abscld 15341	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) ∈ ℝ)
72	69	abscld 15341	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℝ)
73	72	rehalfcld 12363	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ)
74	71, 73, 72	ltsub2d 11722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))))))
75	66, 68	absmuld 15359	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))))
76	64, 67	absexpd 15357	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝑥↑𝑁)) = ((abs‘𝑥)↑𝑁))
77	76	oveq2d 7357	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)))
78	75, 77	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)))
79	78	oveq1d 7356	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2))
80	66	abscld 15341	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℝ)
81	80	recnd 11135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℂ)
82	58	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
83	82, 67	reexpcld 14065	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
84	83	recnd 11135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℂ)
85		2cnd 12198	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ∈ ℂ)
86		2ne0 12224	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ≠ 0)
88	81, 84, 85, 87	div23d 11929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
89	79, 88	eqtrd 2766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
90	89	breq2d 5098	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ↔ (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))))
91	72	recnd 11135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
92	91	2halvesd 12362	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))))
93	92	oveq1d 7356	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)))
94	73	recnd 11135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℂ)
95	94, 94	pncand 11468	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))
96	93, 95	eqtr3d 2768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))
97	96	breq1d 5096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))))))
98	74, 90, 97	3bitr3d 309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))))))
99	69, 65	subcld 11467	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)) ∈ ℂ)
100	69, 99	abs2difd 15362	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) ≤ (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))))
101	69, 65	abssubd 15358	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))))
102	101	oveq2d 7357	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))))
103	69, 65	nncand 11472	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (𝐹‘𝑥))
104	103	fveq2d 6821	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) = (abs‘(𝐹‘𝑥)))
105	100, 102, 104	3brtr3d 5117	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥)))
106	72, 71	resubcld 11540	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∈ ℝ)
107	65	abscld 15341	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
108		ltletr 11200	. . . . . . . . . . . 12 ⊢ ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∈ ℝ ∧ (abs‘(𝐹‘𝑥)) ∈ ℝ) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))))
109	73, 106, 107, 108	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))))
110	105, 109	mpan2d 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))))
111	98, 110	sylbid 240	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))))
112	32	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) ∈ ℝ)
113	20	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ⁺)
114	113	rpred 12929	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ)
115	114, 82	remulcld 11137	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ∈ ℝ)
116	89, 73	eqeltrrd 2832	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
117	35	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ∈ ℝ)
118	41	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 ∈ ℝ)
119		max2 13081	. . . . . . . . . . . . . . . . . 18 ⊢ ((if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)))
120	39, 35, 119	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)))
121	120, 25	breqtrrdi 5128	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ 𝑈)
122	121	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ≤ 𝑈)
123		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 < (abs‘𝑥))
124	117, 118, 82, 122, 123	lelttrd 11266	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
125	26, 124	eqbrtrrid 5122	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥))
126	112, 82, 113	ltdivmuld 12980	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥) ↔ (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥))))
127	125, 126	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)))
128	82	recnd 11135	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
129	128	exp1d 14043	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) = (abs‘𝑥))
130		1red 11108	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ∈ ℝ)
131	51	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ 𝑈)
132	130, 118, 82, 131, 123	lelttrd 11266	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 < (abs‘𝑥))
133	130, 82, 132	ltled 11256	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
134	4	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ)
135		nnuz 12770	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
136	134, 135	eleqtrdi 2841	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘1))
137	82, 133, 136	leexp2ad 14156	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) ≤ ((abs‘𝑥)↑𝑁))
138	129, 137	eqbrtrrd 5110	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁))
139	82, 83, 113	lemul2d 12973	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁) ↔ (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ≤ (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))))
140	138, 139	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ≤ (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
141	112, 115, 116, 127, 140	ltletrd 11268	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
142	141, 89	breqtrrd 5114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))
143		lttr 11184	. . . . . . . . . . 11 ⊢ (((abs‘(𝐹‘0)) ∈ ℝ ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ ∧ (abs‘(𝐹‘𝑥)) ∈ ℝ) → (((abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
144	112, 73, 107, 143	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
145	142, 144	mpand 695	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
146	111, 145	syld 47	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
147	146	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (𝑈 < (abs‘𝑥) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
148	147	a2d 29	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑈 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
149	62, 148	syld 47	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
150	149	ralimdva 3144	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∀𝑥 ∈ ℂ (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
151		breq1 5089	. . . . 5 ⊢ (𝑟 = 𝑈 → (𝑟 < (abs‘𝑥) ↔ 𝑈 < (abs‘𝑥)))
152	151	rspceaimv 3578	. . . 4 ⊢ ((𝑈 ∈ ℝ⁺ ∧ ∀𝑥 ∈ ℂ (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
153	53, 150, 152	syl6an 684	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
154	153	rexlimdva 3133	. 2 ⊢ (𝜑 → (∃𝑠 ∈ ℝ ∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
155	24, 154	mpd 15	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ⁺ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ifcif 4470 class class class wbr 5086 ⟶wf 6472 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 ℝcr 11000 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 < clt 11141 ≤ cle 11142 − cmin 11339 / cdiv 11769 ℕcn 12120 2c2 12175 ℕ₀cn0 12376 ℤ_≥cuz 12727 ℝ⁺crp 12885 ...cfz 13402 ↑cexp 13963 abscabs 15136 Σcsu 15588 0_𝑝c0p 25592 Polycply 26111 coeffccoe 26113 degcdgr 26114

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-pm 8748 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-n0 12377 df-z 12464 df-uz 12728 df-rp 12886 df-ico 13246 df-fz 13403 df-fzo 13550 df-fl 13691 df-seq 13904 df-exp 13964 df-hash 14233 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-clim 15390 df-rlim 15391 df-sum 15589 df-0p 25593 df-ply 26115 df-coe 26117 df-dgr 26118

This theorem is referenced by: fta 27012

W3C validator