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 27290

Description: Lemma for fta 27295. 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 27288; 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 4037	. . 3 ⊢ 𝐷 ⊆ ℂ
3		ftalem3.6	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
4	3	cnfldtopon 24990	. . . . . . 7 ⊢ 𝐽 ∈ (TopOn‘ℂ)
5		resttopon 23368	. . . . . . 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 23123	. . . . 5 ⊢ 𝐷 = ∪ (𝐽 ↾_t 𝐷)
8		eqid 2765	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
9		cnxmet 24980	. . . . . . . 8 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		0cn 11213	. . . . . . . 8 ⊢ 0 ∈ ℂ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℂ)
13		ftalem3.7	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
14	13	rpxrd 13077	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ^*)
15	3	cnfldtopn 24989	. . . . . . . 8 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
16		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) = (abs ∘ − )
17	16	cnmetdval 24978	. . . . . . . . . . . . 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 11459	. . . . . . . . . . . . . 14 ⊢ -𝑦 = (0 − 𝑦)
20	19	fveq2i 6888	. . . . . . . . . . . . 13 ⊢ (abs‘-𝑦) = (abs‘(0 − 𝑦))
21		absneg 15352	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℂ → (abs‘-𝑦) = (abs‘𝑦))
22	20, 21	eqtr3id 2814	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℂ → (abs‘(0 − 𝑦)) = (abs‘𝑦))
23	18, 22	eqtrd 2800	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ → (0(abs ∘ − )𝑦) = (abs‘𝑦))
24	23	breq1d 5121	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℂ → ((0(abs ∘ − )𝑦) ≤ 𝑅 ↔ (abs‘𝑦) ≤ 𝑅))
25	24	rabbiia 3422	. . . . . . . . 9 ⊢ {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅} = {𝑦 ∈ ℂ ∣ (abs‘𝑦) ≤ 𝑅}
26	1, 25	eqtr4i 2791	. . . . . . . 8 ⊢ 𝐷 = {𝑦 ∈ ℂ ∣ (0(abs ∘ − )𝑦) ≤ 𝑅}
27	15, 26	blcld 24713	. . . . . . 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 13076	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℝ)
30		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (abs‘𝑦) = (abs‘𝑥))
31	30	breq1d 5121	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((abs‘𝑦) ≤ 𝑅 ↔ (abs‘𝑥) ≤ 𝑅))
32	31, 1	elrab2 3656	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ℂ ∧ (abs‘𝑥) ≤ 𝑅))
33	32	simprbi 503	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → (abs‘𝑥) ≤ 𝑅)
34	33	rgen 3083	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐷 (abs‘𝑥) ≤ 𝑅
35		brralrspcev 5173	. . . . . . 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 2765	. . . . . . . 8 ⊢ (𝐽 ↾_t 𝐷) = (𝐽 ↾_t 𝐷)
38	3, 37	cnheibor 25165	. . . . . . 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 26469	. . . . . . . . 9 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (ℂ–cn→ℂ))
43	41, 42	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
44		abscncf 25111	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℝ)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝜑 → abs ∈ (ℂ–cn→ℝ))
46	43, 45	cncfco 25117	. . . . . . 7 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (ℂ–cn→ℝ))
47		ssid 3960	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
48		ax-resscn 11172	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
49	4	toponrestid 23128	. . . . . . . . 9 ⊢ 𝐽 = (𝐽 ↾_t ℂ)
50	3	tgioo2 25011	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (𝐽 ↾_t ℝ)
51	3, 49, 50	cncfcn 25120	. . . . . . . 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 2875	. . . . . 6 ⊢ (𝜑 → (abs ∘ 𝐹) ∈ (𝐽 Cn (topGen‘ran (,))))
54	4	toponunii 23123	. . . . . . 7 ⊢ ℂ = ∪ 𝐽
55	54	cnrest 23492	. . . . . 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 13080	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑅)
58		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦 = 0 → (abs‘𝑦) = (abs‘0))
59		abs0 15360	. . . . . . . . . 10 ⊢ (abs‘0) = 0
60	58, 59	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝑦 = 0 → (abs‘𝑦) = 0)
61	60	breq1d 5121	. . . . . . . 8 ⊢ (𝑦 = 0 → ((abs‘𝑦) ≤ 𝑅 ↔ 0 ≤ 𝑅))
62	61, 1	elrab2 3656	. . . . . . 7 ⊢ (0 ∈ 𝐷 ↔ (0 ∈ ℂ ∧ 0 ≤ 𝑅))
63	12, 57, 62	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝐷)
64	63	ne0d 4295	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ ∅)
65	7, 8, 40, 56, 64	evth2 25170	. . . 4 ⊢ (𝜑 → ∃𝑧 ∈ 𝐷 ∀𝑥 ∈ 𝐷 (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥))
66		fvres 6904	. . . . . . . . 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 26406	. . . . . . . . . . 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 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑧 ∈ ℂ)
73		fvco3 6985	. . . . . . . . 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 2800	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) = (abs‘(𝐹‘𝑧)))
76		fvres 6904	. . . . . . . . 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 3936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ ℂ)
80		fvco3 6985	. . . . . . . . 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 2800	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) = (abs‘(𝐹‘𝑥)))
83	75, 82	breq12d 5124	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝐷) → ((((abs ∘ 𝐹) ↾ 𝐷)‘𝑧) ≤ (((abs ∘ 𝐹) ↾ 𝐷)‘𝑥) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥))))
84	83	ralbidva 3188	. . . . 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 3189	. . . 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 4008	. . 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 6890	. . . . . . . . 9 ⊢ (𝑥 = 0 → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘0)))
91	90	breq2d 5123	. . . . . . . 8 ⊢ (𝑥 = 0 → ((abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘𝑥)) ↔ (abs‘(𝐹‘𝑧)) ≤ (abs‘(𝐹‘0))))
92	91	rspcv 3579	. . . . . . 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 7080	. . . . . . . . . . 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 15514	. . . . . . . . 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 3918	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → 𝑥 ∈ ℂ)
100	94, 99	ffvelcdmd 7084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑥) ∈ ℂ)
101	100	abscld 15514	. . . . . . . . 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 3919	. . . . . . . . . . . 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 15514	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘𝑥) ∈ ℝ)
110	108, 109	ltnled 11372	. . . . . . . . . . 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 3255	. . . . . . . . . 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 11373	. . . . . . . 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 7084	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (𝐹‘𝑧) ∈ ℂ)
117	116	abscld 15514	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℂ) ∧ 𝑥 ∈ (ℂ ∖ 𝐷)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
118		letr 11319	. . . . . . . . 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 3167	. . . . . 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 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 2788	. . . . . 6 ⊢ (𝐷 ∪ (ℂ ∖ 𝐷)) = ℂ
129	128	raleqi 3323	. . . . 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 3180	. 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 2146 ∀wral 3081 ∃wrex 3091 {crab 3418 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 class class class wbr 5111 ran crn 5664 ↾ cres 5665 ∘ ccom 5667 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℂcc 11113 ℝcr 11114 0cc0 11115 ℝ^*cxr 11257 < clt 11258 ≤ cle 11259 − cmin 11456 -cneg 11457 ℕcn 12248 ℝ⁺crp 13032 (,)cioo 13388 abscabs 15309 ↾_t crest 17495 TopOpenctopn 17496 topGenctg 17512 ∞Metcxmet 21557 ℂ_fldccnfld 21572 TopOnctopon 23117 Clsdccld 23223 Cn ccn 23431 Compccmp 23593 –cn→ccncf 25086 Polycply 26392 coeffccoe 26394 degcdgr 26395

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ioo 13392 df-icc 13395 df-fz 13552 df-fzo 13700 df-fl 13843 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-rlim 15564 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-rest 17497 df-topn 17498 df-0g 17516 df-gsum 17517 df-topgen 17518 df-pt 17519 df-prds 17522 df-xrs 17578 df-qtop 17583 df-imas 17584 df-xps 17586 df-mre 17660 df-mrc 17661 df-acs 17663 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-mulg 19178 df-cntz 19431 df-cmn 19896 df-psmet 21564 df-xmet 21565 df-met 21566 df-bl 21567 df-mopn 21568 df-cnfld 21573 df-top 23101 df-topon 23118 df-topsp 23140 df-bases 23153 df-cld 23226 df-cls 23228 df-cn 23434 df-cnp 23435 df-haus 23522 df-cmp 23594 df-tx 23770 df-hmeo 23963 df-xms 24528 df-ms 24529 df-tms 24530 df-cncf 25088 df-0p 25880 df-ply 26396 df-coe 26398 df-dgr 26399

This theorem is used by: fta 27295

W3C validator