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Theorem ftalem3 27395

Description: Lemma for fta 27400. 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 27393; 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 4030	. . 3 ⊢ 𝐷 ⊆ ℂ
3		ftalem3.6	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 25094	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
5		resttopon 23472	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝐷 ⊆ ℂ) → (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷))
6	4, 2, 5	mp2an 705	. . . . . 6 ⊢ (𝐽 ↾_t 𝐷) ∈ (TopOn‘𝐷)
7	6	toponunii 23227	. . . . 5 ⊢ 𝐷 = ∪ (𝐽 ↾_t 𝐷)
8		eqid 2761	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
9		cnxmet 25084	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		0cn 11291	. . . . . . . 8 ⊢ 0 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
13		ftalem3.7	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
14	13	rpxrd 13158	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
15	3	cnfldtopn 25093	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
16		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 25082	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
18	11, 17	mpan 703	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘(0 − 𝑦)))
19		df-neg 11537	. . . . . . . . . . . . . 14 ⊢ -𝑦 = (0 − 𝑦)
20	19	fveq2i 6886	. . . . . . . . . . . . 13 ⊢ (abs‘-𝑦) = (abs‘(0 − 𝑦))
21		absneg 15437	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (abs‘-𝑦) = (abs‘𝑦))
22	20, 21	eqtr3id 2810	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (abs‘(0 − 𝑦)) = (abs‘𝑦))
23	18, 22	eqtrd 2796	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘𝑦))
24	23	breq1d 5113	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((0(abs ∘ − )𝑦) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
25	24	rabbiia 3417	. . . . . . . . 9 ⊢ {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
26	1, 25	eqtr4i 2787	. . . . . . . 8 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅}
27	15, 26	blcld 24817	. . . . . . 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 13157	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥))
31	30	breq1d 5113	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
32	31, 1	elrab2 3649	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅))
33	32	simprbi 503	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅)
34	33	rgen 3079	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅
35		brralrspcev 5165	. . . . . . 7 ⊢ ((𝑅 ∈ ℝ ∧ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅) → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
36	29, 34, 35	sylancl 598	. . . . . 6 ⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑠)
37		eqid 2761	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝐷) = (𝐽 ↾_t 𝐷)
38	3, 37	cnheibor 25269	. . . . . . 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 595	. . . . 5 ⊢ (𝜑 → (𝐽 ↾_t 𝐷) ∈ Comp)
41		ftalem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆))
42		plycn 26573	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
43	41, 42	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
44		abscncf 25215	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℝ)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
46	43, 45	cncfco 25221	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ))
47		ssid 3953	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
48		ax-resscn 11250	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
49	4	toponrestid 23232	. . . . . . . . 9 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
50	3	tgioo2 25115	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
51	3, 49, 50	cncfcn 25224	. . . . . . . 8 ⊢ ((ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ) → (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,))))
52	47, 48, 51	mp2an 705	. . . . . . 7 ⊢ (ℂ–cn→ℝ) = (𝐽 Cn (topGen‘ran (,)))
53	46, 52	eleqtrdi 2871	. . . . . 6 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))))
54	4	toponunii 23227	. . . . . . 7 ⊢ ℂ = ∪ 𝐽
55	54	cnrest 23596	. . . . . 6 ⊢ (((abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))) ∧ 𝐷 ⊆ ℂ) → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
56	53, 2, 55	sylancl 598	. . . . 5 ⊢ (𝜑 → ((abs ∘ 𝐹) ↾ 𝐷) ∈ ((𝐽 ↾_t 𝐷) Cn (topGen‘ran (,))))
57	13	rpge0d 13161	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑅)
58		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (abs‘𝑦) = (abs‘0))
59		abs0 15445	. . . . . . . . . 10 ⊢ (abs‘0) = 0
60	58, 59	eqtrdi 2812	. . . . . . . . 9 ⊢ (𝑦 = 0 → (abs‘𝑦) = 0)
61	60	breq1d 5113	. . . . . . . 8 ⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅))
62	61, 1	elrab2 3649	. . . . . . 7 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ 0 ≤ 𝑅))
63	12, 57, 62	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐷)
64	63	ne0d 4288	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
65	7, 8, 40, 56, 64	evth2 25274	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥))
66		fvres 6902	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
67	66	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = ((abs ∘ 𝐹)‘𝑧))
68		plyf 26509	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ)
69	41, 68	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
70	69	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝐹:ℂ⟶ℂ)
71		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ 𝐷)
72	2, 71	sselid 3929	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ)
73		fvco3 6983	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑧 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
74	70, 72, 73	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑧) = (abs‘(𝐹‘𝑧)))
75	67, 74	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧)))
76		fvres 6902	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
77	76	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = ((abs ∘ 𝐹)‘𝑥))
78		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
79	2, 78	sselid 3929	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
80		fvco3 6983	. . . . . . . . 9 ⊢ ((𝐹:ℂ⟶ℂ ∧ 𝑥 ∈ ℂ) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
81	70, 79, 80	syl2anc 596	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((abs ∘ 𝐹)‘𝑥) = (abs‘(𝐹‘𝑥)))
82	77, 81	eqtrd 2796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥)))
83	75, 82	breq12d 5116	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
84	83	ralbidva 3184	. . . . 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 3185	. . . 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 4001	. . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℂ) → 0 ∈ 𝐷)
90		2fveq3 6888	. . . . . . . . 9 ⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0)))
91	90	breq2d 5115	. . . . . . . 8 ⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
92	91	rspcv 3573	. . . . . . 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 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝐹:ℂ⟶ℂ)
95		ffvelcdm 7079	. . . . . . . . . . 11 ⊢ ((𝐹:ℂ⟶ℂ ∧ 0 ∈ ℂ) → (𝐹‘0) ∈ ℂ)
96	94, 11, 95	sylancl 598	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘0) ∈ ℂ)
97	96	abscld 15599	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ∈ ℝ)
98		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ (ℂ ∖ 𝐷))
99	98	eldifad 3911	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ)
100	94, 99	ffvelcdmd 7083	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15599	. . . . . . . . 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 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ∀𝑥 ∈ ℂ (𝑅 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))
104	98	eldifbd 3912	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ¬ 𝑥 ∈ 𝐷)
105	32	baib 545	. . . . . . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑅 ∈ ℝ)
109	99	abscld 15599	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ)
110	108, 109	ltnled 11450	. . . . . . . . . . 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 3251	. . . . . . . . . 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 11451	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘0)) ≤ (abs‘(𝐹‘𝑥)))
115		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑧 ∈ ℂ)
116	94, 115	ffvelcdmd 7083	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ)
117	116	abscld 15599	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
118		letr 11397	. . . . . . . . 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 707	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0)) → (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
121	120	ralrimdva 3163	. . . . . 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 560	. . . 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 4143	. . . . 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 4431	. . . . . . 7 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = (𝐷 ∪ ℂ)
126		ssequn1 4132	. . . . . . . 8 ⊢ (𝐷 ⊆ ℂ ↔ (𝐷 ∪ ℂ) = ℂ)
127	2, 126	mpbi 233	. . . . . . 7 ⊢ (𝐷 ∪ ℂ) = ℂ
128	125, 127	eqtri 2784	. . . . . 6 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ
129	128	raleqi 3318	. . . . 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 3176	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 {crab 3413 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5103 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 0cc0 11193 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 -cneg 11535 ℕcn 12328 ℝ⁺crp 13113 (,)cioo 13469 abscabs 15394 ↾_t crest 17584 TopOpenctopn 17585 topGenctg 17601 ∞Metcxmet 21656 ℂ_fldccnfld 21671 TopOnctopon 23221 Clsdccld 23327 Cn ccn 23535 Compccmp 23697 –cn→ccncf 25190 Polycply 26495 coeffccoe 26497 degcdgr 26498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-icc 13476 df-fz 13633 df-fzo 13782 df-fl 13925 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-rlim 15649 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-cls 23332 df-cn 23538 df-cnp 23539 df-haus 23626 df-cmp 23698 df-tx 23874 df-hmeo 24067 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-0p 25984 df-ply 26499 df-coe 26501 df-dgr 26502

This theorem is used by: fta 27400

W3C validator