ftalem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ftalem3		Structured version Visualization version GIF version

Theorem ftalem3 26992

Description: Lemma for fta 26997. 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 26990; 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 4048	. . 3 ⊢ 𝐷 ⊆ ℂ
3		ftalem3.6	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24677	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
5		resttopon 23055	. . . . . . 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 22810	. . . . 5 ⊢ 𝐷 = ∪ (𝐽 ↾_t 𝐷)
8		eqid 2730	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
9		cnxmet 24667	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		0cn 11173	. . . . . . . 8 ⊢ 0 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
13		ftalem3.7	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
14	13	rpxrd 13003	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
15	3	cnfldtopn 24676	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
16		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24665	. . . . . . . . . . . . 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 11415	. . . . . . . . . . . . . 14 ⊢ -𝑦 = (0 − 𝑦)
20	19	fveq2i 6864	. . . . . . . . . . . . 13 ⊢ (abs‘-𝑦) = (abs‘(0 − 𝑦))
21		absneg 15250	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (abs‘-𝑦) = (abs‘𝑦))
22	20, 21	eqtr3id 2779	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (abs‘(0 − 𝑦)) = (abs‘𝑦))
23	18, 22	eqtrd 2765	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘𝑦))
24	23	breq1d 5120	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((0(abs ∘ − )𝑦) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
25	24	rabbiia 3412	. . . . . . . . 9 ⊢ {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
26	1, 25	eqtr4i 2756	. . . . . . . 8 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅}
27	15, 26	blcld 24400	. . . . . . 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 13002	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥))
31	30	breq1d 5120	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
32	31, 1	elrab2 3665	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅))
33	32	simprbi 496	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅)
34	33	rgen 3047	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅
35		brralrspcev 5170	. . . . . . 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 2730	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝐷) = (𝐽 ↾_t 𝐷)
38	3, 37	cnheibor 24861	. . . . . . 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 26173	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
43	41, 42	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
44		abscncf 24801	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℝ)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
46	43, 45	cncfco 24807	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ))
47		ssid 3972	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
48		ax-resscn 11132	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
49	4	toponrestid 22815	. . . . . . . . 9 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
50	3	tgioo2 24698	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
51	3, 49, 50	cncfcn 24810	. . . . . . . 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 2839	. . . . . 6 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))))
54	4	toponunii 22810	. . . . . . 7 ⊢ ℂ = ∪ 𝐽
55	54	cnrest 23179	. . . . . 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 13006	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑅)
58		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (abs‘𝑦) = (abs‘0))
59		abs0 15258	. . . . . . . . . 10 ⊢ (abs‘0) = 0
60	58, 59	eqtrdi 2781	. . . . . . . . 9 ⊢ (𝑦 = 0 → (abs‘𝑦) = 0)
61	60	breq1d 5120	. . . . . . . 8 ⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅))
62	61, 1	elrab2 3665	. . . . . . 7 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ 0 ≤ 𝑅))
63	12, 57, 62	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐷)
64	63	ne0d 4308	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
65	7, 8, 40, 56, 64	evth2 24866	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥))
66		fvres 6880	. . . . . . . . 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 26110	. . . . . . . . . . 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 3947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ)
73		fvco3 6963	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧)))
76		fvres 6880	. . . . . . . . 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 3947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
80		fvco3 6963	. . . . . . . . 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 2765	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥)))
83	75, 82	breq12d 5123	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
84	83	ralbidva 3155	. . . . 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 3156	. . . 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 4019	. . 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 6866	. . . . . . . . 9 ⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0)))
91	90	breq2d 5122	. . . . . . . 8 ⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
92	91	rspcv 3587	. . . . . . 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 7056	. . . . . . . . . . 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 15412	. . . . . . . . 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 3929	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ)
100	94, 99	ffvelcdmd 7060	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15412	. . . . . . . . 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 3930	. . . . . . . . . . . 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 15412	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ)
110	108, 109	ltnled 11328	. . . . . . . . . . 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 3226	. . . . . . . . . 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 11329	. . . . . . . 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 7060	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ)
117	116	abscld 15412	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
118		letr 11275	. . . . . . . . 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 3134	. . . . . 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 4163	. . . . 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 4443	. . . . . . 7 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = (𝐷 ∪ ℂ)
126		ssequn1 4152	. . . . . . . 8 ⊢ (𝐷 ⊆ ℂ ↔ (𝐷 ∪ ℂ) = ℂ)
127	2, 126	mpbi 230	. . . . . . 7 ⊢ (𝐷 ∪ ℂ) = ℂ
128	125, 127	eqtri 2753	. . . . . 6 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ
129	128	raleqi 3299	. . . . 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 3147	. 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 1540 ∈ wcel 2109 ∀wral 3045 ∃wrex 3054 {crab 3408 ∖ cdif 3914 ∪ cun 3915 ⊆ wss 3917 class class class wbr 5110 ran crn 5642 ↾ cres 5643 ∘ ccom 5645 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 − cmin 11412 -cneg 11413 ℕcn 12193 ℝ⁺crp 12958 (,)cioo 13313 abscabs 15207 ↾_t crest 17390 TopOpenctopn 17391 topGenctg 17407 ∞Metcxmet 21256 ℂ_fldccnfld 21271 TopOnctopon 22804 Clsdccld 22910 Cn ccn 23118 Compccmp 23280 –cn→ccncf 24776 Polycply 26096 coeffccoe 26098 degcdgr 26099

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462 df-sum 15660 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-cls 22915 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-0p 25578 df-ply 26100 df-coe 26102 df-dgr 26103

This theorem is referenced by: fta 26997

W3C validator