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Theorem ftalem3 27239

Description: Lemma for fta 27244. There exists a global minimum of the function abs ∘ 𝐹. The proof uses a circle of radius 𝑟 where 𝑟 is the value coming from ftalem1 27237; since this is a compact set, the minimum on this disk is achieved, and this must then be the global minimum. (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)
ftalem3.5	⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
ftalem3.6	⊢ 𝐽 = (TopOpen‘ℂ_fld)
ftalem3.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
ftalem3.8	⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))

**Assertion**
Ref	Expression
ftalem3	⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑧,𝐷 𝑥,𝑁 𝑥,𝑦,𝐹,𝑧 𝑥,𝐽,𝑧 𝜑,𝑥,𝑦,𝑧 𝑥,𝑅,𝑦 Allowed substitution hints: 𝐴(𝑦,𝑧) 𝐷(𝑦) 𝑅(𝑧) 𝑆(𝑥,𝑦,𝑧) 𝐽(𝑦) 𝑁(𝑦,𝑧)

Proof of Theorem ftalem3 Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftalem3.5	. . . 4 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
2	1	ssrab3 4036	. . 3 ⊢ 𝐷 ⊆ ℂ
3		ftalem3.6	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24939	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
5		resttopon 23318	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
6	4, 2, 5	mp2an 704	. . . . . 6 ⊢ (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷)
7	6	toponunii 23073	. . . . 5 ⊢ 𝐷 = ∪ (𝐽 ↾_t 𝐷)
8		eqid 2763	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
9		cnxmet 24929	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		0cn 11193	. . . . . . . 8 ⊢ 0 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
13		ftalem3.7	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
14	13	rpxrd 13056	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
15	3	cnfldtopn 24938	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
16		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24927	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
18	11, 17	mpan 702	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
19		df-neg 11439	. . . . . . . . . . . . . 14 ⊢ -𝑦 = (0 − 𝑦)
20	19	fveq2i 6884	. . . . . . . . . . . . 13 ⊢ (abs‘-𝑦) = (abs‘(0 − 𝑦))
21		absneg 15324	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (abs‘-𝑦) = (abs‘𝑦))
22	20, 21	eqtr3id 2812	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (abs‘(0 − 𝑦)) = (abs‘𝑦))
23	18, 22	eqtrd 2798	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘𝑦))
24	23	breq1d 5119	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((0(abs ∘ − )𝑦) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
25	24	rabbiia 3420	. . . . . . . . 9 ⊢ {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
26	1, 25	eqtr4i 2789	. . . . . . . 8 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅}
27	15, 26	blcld 24662	. . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → 𝐷 ∈ (Clsd‘𝐽))
28	10, 12, 14, 27	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
29	13	rpred 13055	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥))
31	30	breq1d 5119	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
32	31, 1	elrab2 3654	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅))
33	32	simprbi 502	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅)
34	33	rgen 3081	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅
35		brralrspcev 5171	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅) → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
36	29, 34, 35	sylancl 597	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
37		eqid 2763	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝐷) = (𝐽 ↾_t 𝐷)
38	3, 37	cnheibor 25114	. . . . . . 7 ⊢ (𝐷 ⊆ ℂ → ((𝐽 ↾_t 𝐷) ∈ Comp ↔ (𝐷 ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)))
39	2, 38	ax-mp 5	. . . . . 6 ⊢ ((𝐽 ↾_t 𝐷) ∈ Comp ↔ (𝐷 ∈ (Clsd‘𝐽) ∧ ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠))
40	28, 36, 39	sylanbrc 594	. . . . 5 ⊢ (𝜑 → (𝐽 ↾_t 𝐷) ∈ Comp)
41		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
42		plycn 26418	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
43	41, 42	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
44		abscncf 25060	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℝ)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
46	43, 45	cncfco 25066	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ))
47		ssid 3959	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
48		ax-resscn 11152	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
49	4	toponrestid 23078	. . . . . . . . 9 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
50	3	tgioo2 24960	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
51	3, 49, 50	cncfcn 25069	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,))))
52	47, 48, 51	mp2an 704	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,)))
53	46, 52	eleqtrdi 2873	. . . . . 6 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))))
54	4	toponunii 23073	. . . . . . 7 ⊢ ℂ = ∪ 𝐽
55	54	cnrest 23442	. . . . . 6 ⊢ (((abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
56	53, 2, 55	sylancl 597	. . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
57	13	rpge0d 13059	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑅)
58		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (abs‘𝑦) = (abs‘0))
59		abs0 15332	. . . . . . . . . 10 ⊢ (abs‘0) = 0
60	58, 59	eqtrdi 2814	. . . . . . . . 9 ⊢ (𝑦 = 0 → (abs‘𝑦) = 0)
61	60	breq1d 5119	. . . . . . . 8 ⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅))
62	61, 1	elrab2 3654	. . . . . . 7 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ 0 ≤ 𝑅))
63	12, 57, 62	sylanbrc 594	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐷)
64	63	ne0d 4295	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
65	7, 8, 40, 56, 64	evth2 25119	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥))
66		fvres 6900	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
67	66	ad2antlr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
68		plyf 26355	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
69	41, 68	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
70	69	ad2antrr 738	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝐹:ℂ⟶ℂ)
71		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ 𝐷)
72	2, 71	sselid 3935	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ)
73		fvco3 6981	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑧 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
74	70, 72, 73	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
75	67, 74	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧)))
76		fvres 6900	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
77	76	adantl 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
78		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
79	2, 78	sselid 3935	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
80		fvco3 6981	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑥 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
81	70, 79, 80	syl2anc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
82	77, 81	eqtrd 2798	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥)))
83	75, 82	breq12d 5122	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
84	83	ralbidva 3186	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
85	84	rexbidva 3187	. . . 4 ⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
86	65, 85	mpbid 235	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
87		ssrexv 4007	. . 3 ⊢ (𝐷 ⊆ ℂ → (∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
88	2, 86, 87	mpsyl 69	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
89	63	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 ∈ 𝐷)
90		2fveq3 6886	. . . . . . . . 9 ⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0)))
91	90	breq2d 5121	. . . . . . . 8 ⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
92	91	rspcv 3577	. . . . . . 7 ⊢ (0 ∈ 𝐷 → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
93	89, 92	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
94	69	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝐹:ℂ⟶ℂ)
95		ffvelcdm 7076	. . . . . . . . . . 11 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
96	94, 11, 95	sylancl 597	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘0) ∈ ℂ)
97	96	abscld 15486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ∈ ℝ)
98		simpr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ (ℂ ∖ 𝐷))
99	98	eldifad 3917	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ)
100	94, 99	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑥)) ∈ ℝ)
102		ftalem3.8	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
103	102	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
104	98	eldifbd 3918	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ 𝑥 ∈ 𝐷)
105	32	baib 544	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅))
106	99, 105	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅))
107	104, 106	mtbid 327	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ (abs‘𝑥) ≤ 𝑅)
108	29	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ)
109	99	abscld 15486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ)
110	108, 109	ltnled 11352	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑅 < (abs‘𝑥) ↔ ¬ (abs‘𝑥) ≤ 𝑅))
111	107, 110	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 < (abs‘𝑥))
112		rsp 3253	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))) → (𝑥 ∈ ℂ → (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))))
113	103, 99, 111, 112	syl3c 67	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))
114	97, 101, 113	ltled 11353	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥)))
115		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑧 ∈ ℂ)
116	94, 115	ffvelcdmd 7080	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ)
117	116	abscld 15486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
118		letr 11299	. . . . . . . . 9 ⊢ (((abs‘(𝐹‘𝑧)) ∈ ℝ ∧ (abs‘(𝐹‘0)) ∈ ℝ ∧ (abs‘(𝐹‘𝑥)) ∈ ℝ) → (((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) ∧ (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
119	117, 97, 101, 118	syl3anc 1398	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) ∧ (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
120	114, 119	mpan2d 706	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
121	120	ralrimdva 3165	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
122	93, 121	syld 48	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
123	122	ancld 559	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))))
124		ralunb 4150	. . . . 5 ⊢ (∀𝑥 ∈ (𝐷 ∪ (ℂ ∖ 𝐷))(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
125		undif2 4438	. . . . . . 7 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = (𝐷 ∪ ℂ)
126		ssequn1 4139	. . . . . . . 8 ⊢ (𝐷 ⊆ ℂ ↔ (𝐷 ∪ ℂ) = ℂ)
127	2, 126	mpbi 233	. . . . . . 7 ⊢ (𝐷 ∪ ℂ) = ℂ
128	125, 127	eqtri 2786	. . . . . 6 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ
129	128	raleqi 3321	. . . . 5 ⊢ (∀𝑥 ∈ (𝐷 ∪ (ℂ ∖ 𝐷))(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
130	124, 129	bitr3i 280	. . . 4 ⊢ ((∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ∧ ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))) ↔ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
131	123, 130	imbitrdi 254	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
132	131	reximdva 3178	. 2 ⊢ (𝜑 → (∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
133	88, 132	mpd 16	1 ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 {crab 3416 ∖ cdif 3902 ∪ cun 3903 ⊆ wss 3905 class class class wbr 5109 ran crn 5662 ↾ cres 5663 ∘ ccom 5665 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℂcc 11093 ℝcr 11094 0cc0 11095 ℝ^*cxr 11237 < clt 11238 ≤ cle 11239 − cmin 11436 -cneg 11437 ℕcn 12228 ℝ⁺crp 13011 (,)cioo 13367 abscabs 15281 ↾_t crest 17468 TopOpenctopn 17469 topGenctg 17485 ∞Metcxmet 21507 ℂ_fldccnfld 21522 TopOnctopon 23067 Clsdccld 23173 Cn ccn 23381 Compccmp 23543 –cn→ccncf 25035 Polycply 26341 coeffccoe 26343 degcdgr 26344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12858 df-q 12968 df-rp 13012 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13371 df-icc 13374 df-fz 13531 df-fzo 13679 df-fl 13821 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-rlim 15536 df-sum 15734 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-hom 17329 df-cco 17330 df-rest 17470 df-topn 17471 df-0g 17489 df-gsum 17490 df-topgen 17491 df-pt 17492 df-prds 17495 df-xrs 17551 df-qtop 17556 df-imas 17557 df-xps 17559 df-mre 17633 df-mrc 17634 df-acs 17636 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-mulg 19129 df-cntz 19382 df-cmn 19847 df-psmet 21514 df-xmet 21515 df-met 21516 df-bl 21517 df-mopn 21518 df-cnfld 21523 df-top 23051 df-topon 23068 df-topsp 23090 df-bases 23103 df-cld 23176 df-cls 23178 df-cn 23384 df-cnp 23385 df-haus 23472 df-cmp 23544 df-tx 23719 df-hmeo 23912 df-xms 24477 df-ms 24478 df-tms 24479 df-cncf 25037 df-0p 25829 df-ply 26345 df-coe 26347 df-dgr 26348

This theorem is referenced by: fta 27244

W3C validator