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Theorem ftalem3 27012

Description: Lemma for fta 27017. 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 27010; 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 4029	. . 3 ⊢ 𝐷 ⊆ ℂ
3		ftalem3.6	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24697	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
5		resttopon 23076	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
6	4, 2, 5	mp2an 692	. . . . . 6 ⊢ (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷)
7	6	toponunii 22831	. . . . 5 ⊢ 𝐷 = ∪ (𝐽 ↾_t 𝐷)
8		eqid 2731	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
9		cnxmet 24687	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		0cn 11104	. . . . . . . 8 ⊢ 0 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
13		ftalem3.7	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
14	13	rpxrd 12935	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
15	3	cnfldtopn 24696	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
16		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24685	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
18	11, 17	mpan 690	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
19		df-neg 11347	. . . . . . . . . . . . . 14 ⊢ -𝑦 = (0 − 𝑦)
20	19	fveq2i 6825	. . . . . . . . . . . . 13 ⊢ (abs‘-𝑦) = (abs‘(0 − 𝑦))
21		absneg 15184	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (abs‘-𝑦) = (abs‘𝑦))
22	20, 21	eqtr3id 2780	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (abs‘(0 − 𝑦)) = (abs‘𝑦))
23	18, 22	eqtrd 2766	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘𝑦))
24	23	breq1d 5099	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((0(abs ∘ − )𝑦) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
25	24	rabbiia 3399	. . . . . . . . 9 ⊢ {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
26	1, 25	eqtr4i 2757	. . . . . . . 8 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅}
27	15, 26	blcld 24420	. . . . . . 7 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ 𝑅 ∈ ℝ^*) → 𝐷 ∈ (Clsd‘𝐽))
28	10, 12, 14, 27	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
29	13	rpred 12934	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥))
31	30	breq1d 5099	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
32	31, 1	elrab2 3645	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅))
33	32	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅)
34	33	rgen 3049	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅
35		brralrspcev 5149	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅) → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
36	29, 34, 35	sylancl 586	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
37		eqid 2731	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝐷) = (𝐽 ↾_t 𝐷)
38	3, 37	cnheibor 24881	. . . . . . 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 583	. . . . 5 ⊢ (𝜑 → (𝐽 ↾_t 𝐷) ∈ Comp)
41		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
42		plycn 26193	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
43	41, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
44		abscncf 24821	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℝ)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
46	43, 45	cncfco 24827	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ))
47		ssid 3952	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
48		ax-resscn 11063	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
49	4	toponrestid 22836	. . . . . . . . 9 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
50	3	tgioo2 24718	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
51	3, 49, 50	cncfcn 24830	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,))))
52	47, 48, 51	mp2an 692	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,)))
53	46, 52	eleqtrdi 2841	. . . . . 6 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))))
54	4	toponunii 22831	. . . . . . 7 ⊢ ℂ = ∪ 𝐽
55	54	cnrest 23200	. . . . . 6 ⊢ (((abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
56	53, 2, 55	sylancl 586	. . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
57	13	rpge0d 12938	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑅)
58		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (abs‘𝑦) = (abs‘0))
59		abs0 15192	. . . . . . . . . 10 ⊢ (abs‘0) = 0
60	58, 59	eqtrdi 2782	. . . . . . . . 9 ⊢ (𝑦 = 0 → (abs‘𝑦) = 0)
61	60	breq1d 5099	. . . . . . . 8 ⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅))
62	61, 1	elrab2 3645	. . . . . . 7 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ 0 ≤ 𝑅))
63	12, 57, 62	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐷)
64	63	ne0d 4289	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
65	7, 8, 40, 56, 64	evth2 24886	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥))
66		fvres 6841	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
67	66	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
68		plyf 26130	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
69	41, 68	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
70	69	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝐹:ℂ⟶ℂ)
71		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ 𝐷)
72	2, 71	sselid 3927	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ)
73		fvco3 6921	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑧 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
74	70, 72, 73	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
75	67, 74	eqtrd 2766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧)))
76		fvres 6841	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
77	76	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
78		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
79	2, 78	sselid 3927	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
80		fvco3 6921	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑥 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
81	70, 79, 80	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
82	77, 81	eqtrd 2766	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥)))
83	75, 82	breq12d 5102	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
84	83	ralbidva 3153	. . . . 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 3154	. . . 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 232	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
87		ssrexv 3999	. . 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 68	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℂ ∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
89	63	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 ∈ 𝐷)
90		2fveq3 6827	. . . . . . . . 9 ⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0)))
91	90	breq2d 5101	. . . . . . . 8 ⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
92	91	rspcv 3568	. . . . . . 7 ⊢ (0 ∈ 𝐷 → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
93	89, 92	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
94	69	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝐹:ℂ⟶ℂ)
95		ffvelcdm 7014	. . . . . . . . . . 11 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
96	94, 11, 95	sylancl 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘0) ∈ ℂ)
97	96	abscld 15346	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ∈ ℝ)
98		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ (ℂ ∖ 𝐷))
99	98	eldifad 3909	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ)
100	94, 99	ffvelcdmd 7018	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15346	. . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
104	98	eldifbd 3910	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ 𝑥 ∈ 𝐷)
105	32	baib 535	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅))
106	99, 105	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑥 ∈ 𝐷 ↔ (abs‘𝑥) ≤ 𝑅))
107	104, 106	mtbid 324	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ (abs‘𝑥) ≤ 𝑅)
108	29	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ)
109	99	abscld 15346	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ)
110	108, 109	ltnled 11260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝑅 < (abs‘𝑥) ↔ ¬ (abs‘𝑥) ≤ 𝑅))
111	107, 110	mpbird 257	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 < (abs‘𝑥))
112		rsp 3220	. . . . . . . . . 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 66	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))
114	97, 101, 113	ltled 11261	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥)))
115		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑧 ∈ ℂ)
116	94, 115	ffvelcdmd 7018	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ)
117	116	abscld 15346	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
118		letr 11207	. . . . . . . . 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 1373	. . . . . . . 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 694	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
121	120	ralrimdva 3132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
122	93, 121	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ (ℂ ∖ 𝐷)(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
123	122	ancld 550	. . . 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 4144	. . . . 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 4424	. . . . . . 7 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = (𝐷 ∪ ℂ)
126		ssequn1 4133	. . . . . . . 8 ⊢ (𝐷 ⊆ ℂ ↔ (𝐷 ∪ ℂ) = ℂ)
127	2, 126	mpbi 230	. . . . . . 7 ⊢ (𝐷 ∪ ℂ) = ℂ
128	125, 127	eqtri 2754	. . . . . 6 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ
129	128	raleqi 3290	. . . . 5 ⊢ (∀𝑥 ∈ (𝐷 ∪ (ℂ ∖ 𝐷))(abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)))
130	124, 129	bitr3i 277	. . . 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 251	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → (∀𝑥 ∈ 𝐷 (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) → ∀𝑥 ∈ ℂ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
132	131	reximdva 3145	. 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 15	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 {crab 3395 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 class class class wbr 5089 ran crn 5615 ↾ cres 5616 ∘ ccom 5618 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 -cneg 11345 ℕcn 12125 ℝ⁺crp 12890 (,)cioo 13245 abscabs 15141 ↾_t crest 17324 TopOpenctopn 17325 topGenctg 17341 ∞Metcxmet 21276 ℂ_fldccnfld 21291 TopOnctopon 22825 Clsdccld 22931 Cn ccn 23139 Compccmp 23301 –cn→ccncf 24796 Polycply 26116 coeffccoe 26118 degcdgr 26119

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-cls 22936 df-cn 23142 df-cnp 23143 df-haus 23230 df-cmp 23302 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-0p 25598 df-ply 26120 df-coe 26122 df-dgr 26123

This theorem is referenced by: fta 27017

W3C validator