ftalem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ftalem2		Structured version Visualization version GIF version

Theorem ftalem2 26984

Description: Lemma for fta 26990. 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 26137	. . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ₀⟶ℂ)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴:ℕ₀⟶ℂ)
7	4	nnnn0d 12503	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
8	6, 7	ffvelcdmd 7057	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ∈ ℂ)
9	4	nnne0d 12236	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
10	2, 1	dgreq0 26171	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0_𝑝 ↔ (𝐴‘𝑁) = 0))
11		fveq2 6858	. . . . . . . . . . 11 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = (deg‘0_𝑝))
12		dgr0 26168	. . . . . . . . . . 11 ⊢ (deg‘0_𝑝) = 0
13	11, 12	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝐹 = 0_𝑝 → (deg‘𝐹) = 0)
14	2, 13	eqtrid 2776	. . . . . . . . 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 2946	. . . . . 6 ⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0))
18	9, 17	mpd 15	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑁) ≠ 0)
19	8, 18	absrpcld 15417	. . . 4 ⊢ (𝜑 → (abs‘(𝐴‘𝑁)) ∈ ℝ⁺)
20	19	rphalfcld 13007	. . 3 ⊢ (𝜑 → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ⁺)
21		2fveq3 6863	. . . . 5 ⊢ (𝑛 = 𝑘 → (abs‘(𝐴‘𝑛)) = (abs‘(𝐴‘𝑘)))
22	21	cbvsumv 15662	. . . 4 ⊢ Σ𝑛 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑛)) = Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘))
23	22	oveq1i 7397	. . 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 26983	. 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 26103	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
28	3, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
29		0cn 11166	. . . . . . . . . . . 12 ⊢ 0 ∈ ℂ
30		ffvelcdm 7053	. . . . . . . . . . . 12 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
31	28, 29, 30	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘0) ∈ ℂ)
32	31	abscld 15405	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘(𝐹‘0)) ∈ ℝ)
33	32, 20	rerpdivcld 13026	. . . . . . . . 9 ⊢ (𝜑 → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ∈ ℝ)
34	26, 33	eqeltrid 2832	. . . . . . . 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 11174	. . . . . . . 8 ⊢ 1 ∈ ℝ
38		ifcl 4534	. . . . . . . 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 4535	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) ∈ ℝ)
41	25, 40	eqeltrid 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ)
42		0red 11177	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈ ℝ)
43		1red 11175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
44		0lt1 11700	. . . . . . 7 ⊢ 0 < 1
45	44	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 1)
46		max1 13145	. . . . . . . 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 13145	. . . . . . . . 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 5149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑈)
51	43, 39, 41, 47, 50	letrd 11331	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ 𝑈)
52	42, 43, 41, 45, 51	ltletrd 11334	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 𝑈)
53	41, 52	elrpd 12992	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ⁺)
54		max2 13147	. . . . . . . . . . 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 11331	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ 𝑈)
57	56	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ≤ 𝑈)
58		abscl 15244	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → (abs‘𝑥) ∈ ℝ)
59		lelttr 11264	. . . . . . . . 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 7057	. . . . . . . . . . . . . 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 14111	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ)
69	66, 68	mulcld 11194	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ)
70	65, 69	subcld 11533	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
71	70	abscld 15405	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) ∈ ℝ)
72	69	abscld 15405	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℝ)
73	72	rehalfcld 12429	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ)
74	71, 73, 72	ltsub2d 11788	. . . . . . . . . . 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 15423	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))))
76	64, 67	absexpd 15421	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝑥↑𝑁)) = ((abs‘𝑥)↑𝑁))
77	76	oveq2d 7403	. . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)))
79	78	oveq1d 7402	. . . . . . . . . . . . 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 15405	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℝ)
81	80	recnd 11202	. . . . . . . . . . . . . 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 14128	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ)
84	83	recnd 11202	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℂ)
85		2cnd 12264	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ∈ ℂ)
86		2ne0 12290	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
87	86	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ≠ 0)
88	81, 84, 85, 87	div23d 11995	. . . . . . . . . . . . 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 2764	. . . . . . . . . . . 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 5119	. . . . . . . . . . 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 11202	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ)
92	91	2halvesd 12428	. . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . 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 11202	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℂ)
95	94, 94	pncand 11534	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 5117	. . . . . . . . . . 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 11533	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)) ∈ ℂ)
100	69, 99	abs2difd 15426	. . . . . . . . . . . 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 15422	. . . . . . . . . . . . 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 7403	. . . . . . . . . . . 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 11538	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (𝐹‘𝑥))
104	103	fveq2d 6862	. . . . . . . . . . . 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 5138	. . . . . . . . . . 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 11606	. . . . . . . . . . . 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 15405	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
108		ltletr 11266	. . . . . . . . . . . 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 12995	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ)
115	114, 82	remulcld 11204	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ∈ ℝ)
116	89, 73	eqeltrrd 2829	. . . . . . . . . . . 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 13147	. . . . . . . . . . . . . . . . . 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 5149	. . . . . . . . . . . . . . . 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 11332	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 < (abs‘𝑥))
125	26, 124	eqbrtrrid 5143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥))
126	112, 82, 113	ltdivmuld 13046	. . . . . . . . . . . . 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 11202	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ)
129	128	exp1d 14106	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) = (abs‘𝑥))
130		1red 11175	. . . . . . . . . . . . . . . 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 11332	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 < (abs‘𝑥))
133	130, 82, 132	ltled 11322	. . . . . . . . . . . . . . 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 12836	. . . . . . . . . . . . . . . 16 ⊢ ℕ = (ℤ_≥‘1)
136	134, 135	eleqtrdi 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ (ℤ_≥‘1))
137	82, 133, 136	leexp2ad 14219	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) ≤ ((abs‘𝑥)↑𝑁))
138	129, 137	eqbrtrrd 5131	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁))
139	82, 83, 113	lemul2d 13039	. . . . . . . . . . . . 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 11334	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))
142	141, 89	breqtrrd 5135	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))
143		lttr 11250	. . . . . . . . . . 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 3145	. . . 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 5110	. . . . 5 ⊢ (𝑟 = 𝑈 → (𝑟 < (abs‘𝑥) ↔ 𝑈 < (abs‘𝑥)))
152	151	rspceaimv 3594	. . . 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 3134	. 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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ifcif 4488 class class class wbr 5107 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 ℕ₀cn0 12442 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 ↑cexp 14026 abscabs 15200 Σcsu 15652 0_𝑝c0p 25570 Polycply 26089 coeffccoe 26091 degcdgr 26092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ico 13312 df-fz 13469 df-fzo 13616 df-fl 13754 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455 df-sum 15653 df-0p 25571 df-ply 26093 df-coe 26095 df-dgr 26096

This theorem is referenced by: fta 26990

W3C validator