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Theorem ftalem2 25659

Description: Lemma for fta 25665. 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 24829	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7	4	nnnn0d 11943	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	6, 7	ffvelrnd 6829	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ∈ ℂ)
9	4	nnne0d 11675	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
10	2, 1	dgreq0 24862	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
11		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
12		dgr0 24859	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
13	11, 12	eqtrdi 2849	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
14	2, 13	syl5eq 2845	. . . . . . . . 9 ⊢ (𝐹 = 0_𝑝 → 𝑁 = 0)
15	10, 14	syl6bir 257	. . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
16	3, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0))
17	16	necon3d 3008	. . . . . 6 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
18	9, 17	mpd 15	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
19	8, 18	absrpcld 14800	. . . 4 ⊢ (𝜑 → (abs‘(𝐴‘𝑁)) ∈ ℝ⁺)
20	19	rphalfcld 12431	. . 3 ⊢ (𝜑 → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ⁺)
21		2fveq3 6650	. . . . 5 ⊢ (𝑛 = 𝑘 → (abs‘(𝐴‘𝑛)) = (abs‘(𝐴‘𝑘)))
22	21	cbvsumv 15045	. . . 4 ⊢ Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑛)) = Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘))
23	22	oveq1i 7145	. . 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 25658	. 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 24795	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
28	3, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
29		0cn 10622	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
30		ffvelrn 6826	. . . . . . . . . . . 12 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
31	28, 29, 30	sylancl 589	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
32	31	abscld 14788	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ)
33	32, 20	rerpdivcld 12450	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ∈ ℝ)
34	26, 33	eqeltrid 2894	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ∈ ℝ)
35	34	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ)
36		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ)
37		1re 10630	. . . . . . . 8 ⊢ 1 ∈ ℝ
38		ifcl 4469	. . . . . . . 8 ⊢ ((𝑠 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ)
39	36, 37, 38	sylancl 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ)
40	35, 39	ifcld 4470	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) ∈ ℝ)
41	25, 40	eqeltrid 2894	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ)
42		0red 10633	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
43		1red 10631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
44		0lt1 11151	. . . . . . 7 ⊢ 0 < 1
45	44	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 1)
46		max1 12566	. . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1))
47	37, 36, 46	sylancr 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1))
48		max1 12566	. . . . . . . . 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 587	. . . . . . . 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 5072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑈)
51	43, 39, 41, 47, 50	letrd 10786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ 𝑈)
52	42, 43, 41, 45, 51	ltletrd 10789	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 𝑈)
53	41, 52	elrpd 12416	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ⁺)
54		max2 12568	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1))
55	37, 36, 54	sylancr 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1))
56	36, 39, 41, 55, 50	letrd 10786	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ 𝑈)
57	56	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ≤ 𝑈)
58		abscl 14630	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
59		lelttr 10720	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (abs‘𝑥) ∈ ℝ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥)))
60	36, 41, 58, 59	syl2an3an 1419	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥)))
61	57, 60	mpand 694	. . . . . . 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 725	. . . . . . . . . . . . . . 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	ffvelrnd 6829	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐹‘𝑥) ∈ ℂ)
66	8	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ)
67	7	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ₀)
68	64, 67	expcld 13506	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
69	66, 68	mulcld 10650	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
70	65, 69	subcld 10986	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
71	70	abscld 14788	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) ∈ ℝ)
72	69	abscld 14788	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℝ)
73	72	rehalfcld 11872	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ)
74	71, 73, 72	ltsub2d 11239	. . . . . . . . . . 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 14806	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))))
76	64, 67	absexpd 14804	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝑥↑𝑁)) = ((abs‘𝑥)↑𝑁))
77	76	oveq2d 7151	. . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)))
79	78	oveq1d 7150	. . . . . . . . . . . . 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 14788	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℝ)
81	80	recnd 10658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℂ)
82	58	ad2antrl 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ)
83	82, 67	reexpcld 13523	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
84	83	recnd 10658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℂ)
85		2cnd 11703	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ∈ ℂ)
86		2ne0 11729	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ≠ 0)
88	81, 84, 85, 87	div23d 11442	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 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 5042	. . . . . . . . . . 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 10658	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
92	91	2halvesd 11871	. . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . 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 10658	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℂ)
95	94, 94	pncand 10987	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 5040	. . . . . . . . . . 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 312	. . . . . . . . . 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 10986	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)) ∈ ℂ)
100	69, 99	abs2difd 14809	. . . . . . . . . . . 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 14805	. . . . . . . . . . . . 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 7151	. . . . . . . . . . . 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 10991	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (𝐹‘𝑥))
104	103	fveq2d 6649	. . . . . . . . . . . 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 5061	. . . . . . . . . . 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 11057	. . . . . . . . . . . 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 14788	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
108		ltletr 10721	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 693	. . . . . . . . . 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 243	. . . . . . . . 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 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) ∈ ℝ)
113	20	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ⁺)
114	113	rpred 12419	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ)
115	114, 82	remulcld 10660	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ∈ ℝ)
116	89, 73	eqeltrrd 2891	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ∈ ℝ)
117	35	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ∈ ℝ)
118	41	adantr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 ∈ ℝ)
119		max2 12568	. . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)))
121	120, 25	breqtrrdi 5072	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ 𝑈)
122	121	adantr 484	. . . . . . . . . . . . . . 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 10787	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
125	26, 124	eqbrtrrid 5066	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥))
126	112, 82, 113	ltdivmuld 12470	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)))
128	82	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
129	128	exp1d 13501	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) = (abs‘𝑥))
130		1red 10631	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ∈ ℝ)
131	51	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ 𝑈)
132	130, 118, 82, 131, 123	lelttrd 10787	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 < (abs‘𝑥))
133	130, 82, 132	ltled 10777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ (abs‘𝑥))
134	4	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ)
135		nnuz 12269	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
136	134, 135	eleqtrdi 2900	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘1))
137	82, 133, 136	leexp2ad 13613	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) ≤ ((abs‘𝑥)↑𝑁))
138	129, 137	eqbrtrrd 5054	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁))
139	82, 83, 113	lemul2d 12463	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 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 10789	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
142	141, 89	breqtrrd 5058	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))
143		lttr 10706	. . . . . . . . . . 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 1368	. . . . . . . . . 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 694	. . . . . . . . 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 460	. . . . . . 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 5033	. . . . 5 ⊢ (𝑟 = 𝑈 → (𝑟 < (abs‘𝑥) ↔ 𝑈 < (abs‘𝑥)))
152	151	rspceaimv 3576	. . . 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 683	. . 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 3243	. 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 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ifcif 4425 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤ_≥cuz 12231 ℝ⁺crp 12377 ...cfz 12885 ↑cexp 13425 abscabs 14585 Σcsu 15034 0_𝑝c0p 24273 Polycply 24781 coeffccoe 24783 degcdgr 24784

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788

This theorem is referenced by: fta 25665

W3C validator