Theorem hlimcaui 5141

Description: If a sequence in Hilbert space subset converges to a limit, it is a Cauchy sequence.

Hypotheses

Ref	Expression
hlimcau.1	⊢ A ∈ V
hlimcau.2	⊢ F ∈ V
hlimcaui.4	⊢ F ⇝v A

Assertion

Ref	Expression
hlimcaui	⊢ F ∈ Cauchy

Proof of Theorem hlimcaui

Step	Hyp	Ref	Expression
1		hlimcaui.4	. . . 4 ⊢ F ⇝v A
2		hlimcau.2	. . . . 5 ⊢ F ∈ V
3		hlimcau.1	. . . . 5 ⊢ A ∈ V
4	2, 3	hlimseq 5109	. . . 4 ⊢ (F ⇝v A → F:ℕ–→ ℋ )
5	1, 4	ax-mp 6	. . 3 ⊢ F:ℕ–→ ℋ
6		breq2 2066	. . . . . . . . 9 ⊢ (v = (x / 2) → (0 < v ↔ 0 < (x / 2)))
7		breq2 2066	. . . . . . . . . . . 12 ⊢ (v = (x / 2) → ((norm ‘((F ‘z) −v A)) < v ↔ (norm ‘((F ‘z) −v A)) < (x / 2)))
8	7	imbi2d 464	. . . . . . . . . . 11 ⊢ (v = (x / 2) → ((y ≤ z → (norm ‘((F ‘z) −v A)) < v) ↔ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
9	8	biraldv 1219	. . . . . . . . . 10 ⊢ (v = (x / 2) → (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
10	9	birexdv 1220	. . . . . . . . 9 ⊢ (v = (x / 2) → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
11	6, 10	imbi12d 474	. . . . . . . 8 ⊢ (v = (x / 2) → ((0 < v → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v)) ↔ (0 < (x / 2) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)))))
12	2, 3	hlimvec 5110	. . . . . . . . . 10 ⊢ (F ⇝v A → A ∈ ℋ )
13	1, 12	ax-mp 6	. . . . . . . . 9 ⊢ A ∈ ℋ
14		hlim2 5112	. . . . . . . . . 10 ⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → (F ⇝v A ↔ ∀v ∈ ℝ (0 < v → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v))))
15	1, 14	mpbii 168	. . . . . . . . 9 ⊢ ((F:ℕ–→ ℋ ∧ A ∈ ℋ ) → ∀v ∈ ℝ (0 < v → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v)))
16	5, 13, 15	mp2an 520	. . . . . . . 8 ⊢ ∀v ∈ ℝ (0 < v → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < v))
17	11, 16	vtoclri 1393	. . . . . . 7 ⊢ ((x / 2) ∈ ℝ → (0 < (x / 2) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
18		opreq1 3006	. . . . . . . . . 10 ⊢ (x = if(x ∈ ℝ, x, 0) → (x / 2) = (if(x ∈ ℝ, x, 0) / 2))
19	18	eleq1d 1155	. . . . . . . . 9 ⊢ (x = if(x ∈ ℝ, x, 0) → ((x / 2) ∈ ℝ ↔ (if(x ∈ ℝ, x, 0) / 2) ∈ ℝ))
20		ax0re 4063	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
21	20	elimel 1793	. . . . . . . . . 10 ⊢ if(x ∈ ℝ, x, 0) ∈ ℝ
22		2re 4470	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
23		2pos 4479	. . . . . . . . . . 11 ⊢ 0 < 2
24	22, 23	gt0ne0i 4345	. . . . . . . . . 10 ⊢ 2 ≠ 0
25	21, 22, 24	redivcl 4274	. . . . . . . . 9 ⊢ (if(x ∈ ℝ, x, 0) / 2) ∈ ℝ
26	19, 25	dedth 1784	. . . . . . . 8 ⊢ (x ∈ ℝ → (x / 2) ∈ ℝ)
27	26	adantr 306	. . . . . . 7 ⊢ ((x ∈ ℝ ∧ 0 < x) → (x / 2) ∈ ℝ)
28		breq2 2066	. . . . . . . . . 10 ⊢ (x = if(x ∈ ℝ, x, 0) → (0 < x ↔ 0 < if(x ∈ ℝ, x, 0)))
29	18	breq2d 2072	. . . . . . . . . 10 ⊢ (x = if(x ∈ ℝ, x, 0) → (0 < (x / 2) ↔ 0 < (if(x ∈ ℝ, x, 0) / 2)))
30	28, 29	imbi12d 474	. . . . . . . . 9 ⊢ (x = if(x ∈ ℝ, x, 0) → ((0 < x → 0 < (x / 2)) ↔ (0 < if(x ∈ ℝ, x, 0) → 0 < (if(x ∈ ℝ, x, 0) / 2))))
31	21, 22, 23	divgt0lem 4389	. . . . . . . . 9 ⊢ (0 < if(x ∈ ℝ, x, 0) → 0 < (if(x ∈ ℝ, x, 0) / 2))
32	30, 31	dedth 1784	. . . . . . . 8 ⊢ (x ∈ ℝ → (0 < x → 0 < (x / 2)))
33	32	imp 277	. . . . . . 7 ⊢ ((x ∈ ℝ ∧ 0 < x) → 0 < (x / 2))
34	17, 27, 33	sylc 62	. . . . . 6 ⊢ ((x ∈ ℝ ∧ 0 < x) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)))
35		prth 429	. . . . . . . . . . . 12 ⊢ (((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) → ((y ≤ z ∧ y ≤ w) → ((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘((F ‘w) −v A)) < (x / 2))))
36		normsubt 5091	. . . . . . . . . . . . . . . . 17 ⊢ ((A ∈ ℋ ∧ (F ‘w) ∈ ℋ ) → (norm ‘(A −v (F ‘w))) = (norm ‘((F ‘w) −v A)))
37	36	breq1d 2071	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ ℋ ∧ (F ‘w) ∈ ℋ ) → ((norm ‘(A −v (F ‘w))) < (x / 2) ↔ (norm ‘((F ‘w) −v A)) < (x / 2)))
38	37	anbi2d 468	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ ℋ ∧ (F ‘w) ∈ ℋ ) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘(A −v (F ‘w))) < (x / 2)) ↔ ((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘((F ‘w) −v A)) < (x / 2))))
39	38	adantl 305	. . . . . . . . . . . . . 14 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ (A ∈ ℋ ∧ (F ‘w) ∈ ℋ )) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘(A −v (F ‘w))) < (x / 2)) ↔ ((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘((F ‘w) −v A)) < (x / 2))))
40		ffvrn 2890	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((F:ℕ–→ ℋ ∧ z ∈ ℕ) → (F ‘z) ∈ ℋ )
41	5, 40	mpan 518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z ∈ ℕ → (F ‘z) ∈ ℋ )
42	41	anim1i 269	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((z ∈ ℕ ∧ (F ‘w) ∈ ℋ ) → ((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ))
43	42	ancoms 334	. . . . . . . . . . . . . . . . . . 19 ⊢ (((F ‘w) ∈ ℋ ∧ z ∈ ℕ) → ((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ))
44	43	anim1i 269	. . . . . . . . . . . . . . . . . 18 ⊢ ((((F ‘w) ∈ ℋ ∧ z ∈ ℕ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)) → (((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)))
45	44	ancoms 334	. . . . . . . . . . . . . . . . 17 ⊢ (((A ∈ ℋ ∧ x ∈ ℝ) ∧ ((F ‘w) ∈ ℋ ∧ z ∈ ℕ)) → (((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)))
46	45	an4s 390	. . . . . . . . . . . . . . . 16 ⊢ (((A ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (x ∈ ℝ ∧ z ∈ ℕ)) → (((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)))
47	46	ancoms 334	. . . . . . . . . . . . . . 15 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ (A ∈ ℋ ∧ (F ‘w) ∈ ℋ )) → (((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)))
48		norm3lemt 5097	. . . . . . . . . . . . . . 15 ⊢ ((((F ‘z) ∈ ℋ ∧ (F ‘w) ∈ ℋ ) ∧ (A ∈ ℋ ∧ x ∈ ℝ)) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘(A −v (F ‘w))) < (x / 2)) → (norm ‘((F ‘z) −v (F ‘w))) < x))
49	47, 48	syl 12	. . . . . . . . . . . . . 14 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ (A ∈ ℋ ∧ (F ‘w) ∈ ℋ )) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘(A −v (F ‘w))) < (x / 2)) → (norm ‘((F ‘z) −v (F ‘w))) < x))
50	39, 49	sylbird 180	. . . . . . . . . . . . 13 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ (A ∈ ℋ ∧ (F ‘w) ∈ ℋ )) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘((F ‘w) −v A)) < (x / 2)) → (norm ‘((F ‘z) −v (F ‘w))) < x))
51		ffvrn 2890	. . . . . . . . . . . . . . 15 ⊢ ((F:ℕ–→ ℋ ∧ w ∈ ℕ) → (F ‘w) ∈ ℋ )
52	5, 51	mpan 518	. . . . . . . . . . . . . 14 ⊢ (w ∈ ℕ → (F ‘w) ∈ ℋ )
53	52, 13	jctil 240	. . . . . . . . . . . . 13 ⊢ (w ∈ ℕ → (A ∈ ℋ ∧ (F ‘w) ∈ ℋ ))
54	50, 53	sylan2 346	. . . . . . . . . . . 12 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ w ∈ ℕ) → (((norm ‘((F ‘z) −v A)) < (x / 2) ∧ (norm ‘((F ‘w) −v A)) < (x / 2)) → (norm ‘((F ‘z) −v (F ‘w))) < x))
55	35, 54	syl9r 56	. . . . . . . . . . 11 ⊢ (((x ∈ ℝ ∧ z ∈ ℕ) ∧ w ∈ ℕ) → (((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) → ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
56	55	r19.20dva 1256	. . . . . . . . . 10 ⊢ ((x ∈ ℝ ∧ z ∈ ℕ) → (∀w ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) → ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
57	56	r19.20dva 1256	. . . . . . . . 9 ⊢ (x ∈ ℝ → (∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) → ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
58		raaan 1775	. . . . . . . . . 10 ⊢ (∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) ↔ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ ∀w ∈ ℕ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))))
59		breq2 2066	. . . . . . . . . . . . 13 ⊢ (w = z → (y ≤ w ↔ y ≤ z))
60		fveq2 2832	. . . . . . . . . . . . . . . 16 ⊢ (w = z → (F ‘w) = (F ‘z))
61	60	opreq1d 3012	. . . . . . . . . . . . . . 15 ⊢ (w = z → ((F ‘w) −v A) = ((F ‘z) −v A))
62	61	fveq2d 2836	. . . . . . . . . . . . . 14 ⊢ (w = z → (norm ‘((F ‘w) −v A)) = (norm ‘((F ‘z) −v A)))
63	62	breq1d 2071	. . . . . . . . . . . . 13 ⊢ (w = z → ((norm ‘((F ‘w) −v A)) < (x / 2) ↔ (norm ‘((F ‘z) −v A)) < (x / 2)))
64	59, 63	imbi12d 474	. . . . . . . . . . . 12 ⊢ (w = z → ((y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2)) ↔ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
65	64	cbvralv 1333	. . . . . . . . . . 11 ⊢ (∀w ∈ ℕ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2)) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)))
66	65	anbi2i 367	. . . . . . . . . 10 ⊢ ((∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ ∀w ∈ ℕ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) ↔ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))))
67		anidm 331	. . . . . . . . . 10 ⊢ ((∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2))) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)))
68	58, 66, 67	3bitr 155	. . . . . . . . 9 ⊢ (∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) ∧ (y ≤ w → (norm ‘((F ‘w) −v A)) < (x / 2))) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)))
69	57, 68	syl5ibr 182	. . . . . . . 8 ⊢ (x ∈ ℝ → (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) → ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
70	69	r19.22sdv 1279	. . . . . . 7 ⊢ (x ∈ ℝ → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
71	70	adantr 306	. . . . . 6 ⊢ ((x ∈ ℝ ∧ 0 < x) → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < (x / 2)) → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
72	34, 71	mpd 46	. . . . 5 ⊢ ((x ∈ ℝ ∧ 0 < x) → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x))
73	72	exp 291	. . . 4 ⊢ (x ∈ ℝ → (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
74	73	rgen 1247	. . 3 ⊢ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x))
75	5, 74	pm3.2i 234	. 2 ⊢ (F:ℕ–→ ℋ ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x)))
76		hcauchy 5103	. 2 ⊢ (F ∈ Cauchy ↔ (F:ℕ–→ ℋ ∧ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ ∀w ∈ ℕ ((y ≤ z ∧ y ≤ w) → (norm ‘((F ‘z) −v (F ‘w))) < x))))
77	75, 76	mpbir 165	1 ⊢ F ∈ Cauchy

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348  ifcif 1776   class class class wbr 2054  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℝcr 4027  0cc0 4028   < clt 4033   / cdiv 4091   ≤ cle 4092  ℕcn 4093  2c2 4454   ℋ chil 4958   −_v cmv 4962  normcno 4964  Cauchyccau 4965   ⇝_v chli 4966

This theorem is referenced by:  hlimcau 5142

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107

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