Theorem hlimunii 5143

Description: A Hilbert space sequence converges to at most one limit.

Hypotheses

Ref	Expression
hlimuni.1	⊢ A ∈ V
hlimuni.2	⊢ B ∈ V
hlimuni.3	⊢ F ∈ V
hlimunii.3	⊢ (F ⇝v A ∧ F ⇝v B)

Assertion

Ref	Expression
hlimunii	⊢ A = B

Proof of Theorem hlimunii

Step	Hyp	Ref	Expression
1		nn2get 4438	. . . . 5 ⊢ ((y ∈ ℕ ∧ w ∈ ℕ) → ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
2	1	rgen2 1248	. . . 4 ⊢ ∀y ∈ ℕ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z)
3		hlimunii.3	. . . . . . . . . 10 ⊢ (F ⇝v A ∧ F ⇝v B)
4	3	pm3.26i 257	. . . . . . . . 9 ⊢ F ⇝v A
5		hlimuni.3	. . . . . . . . . 10 ⊢ F ∈ V
6		hlimuni.1	. . . . . . . . . 10 ⊢ A ∈ V
7	5, 6	hlimvec 5110	. . . . . . . . 9 ⊢ (F ⇝v A → A ∈ ℋ )
8	4, 7	ax-mp 6	. . . . . . . 8 ⊢ A ∈ ℋ
9	3	pm3.27i 261	. . . . . . . . 9 ⊢ F ⇝v B
10		hlimuni.2	. . . . . . . . . 10 ⊢ B ∈ V
11	5, 10	hlimvec 5110	. . . . . . . . 9 ⊢ (F ⇝v B → B ∈ ℋ )
12	9, 11	ax-mp 6	. . . . . . . 8 ⊢ B ∈ ℋ
13	8, 12	hvsubcl 5002	. . . . . . 7 ⊢ (A −v B) ∈ ℋ
14	13	normcl 5081	. . . . . 6 ⊢ (norm ‘(A −v B)) ∈ ℝ
15		2re 4470	. . . . . 6 ⊢ 2 ∈ ℝ
16		2pos 4479	. . . . . 6 ⊢ 0 < 2
17	14, 15, 16	divgt0lem 4389	. . . . 5 ⊢ (0 < (norm ‘(A −v B)) → 0 < ((norm ‘(A −v B)) / 2))
18	5, 6	hlimconv 5111	. . . . . . . 8 ⊢ (F ⇝v A → ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)))
19	4, 18	ax-mp 6	. . . . . . 7 ⊢ ∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x))
20	15, 16	gt0ne0i 4345	. . . . . . . 8 ⊢ 2 ≠ 0
21	14, 15, 20	redivcl 4274	. . . . . . 7 ⊢ ((norm ‘(A −v B)) / 2) ∈ ℝ
22		breq2 2066	. . . . . . . . 9 ⊢ (x = ((norm ‘(A −v B)) / 2) → (0 < x ↔ 0 < ((norm ‘(A −v B)) / 2)))
23		breq2 2066	. . . . . . . . . . . 12 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((norm ‘((F ‘z) −v A)) < x ↔ (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)))
24	23	imbi2d 464	. . . . . . . . . . 11 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((y ≤ z → (norm ‘((F ‘z) −v A)) < x) ↔ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2))))
25	24	biraldv 1219	. . . . . . . . . 10 ⊢ (x = ((norm ‘(A −v B)) / 2) → (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x) ↔ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2))))
26	25	birexdv 1220	. . . . . . . . 9 ⊢ (x = ((norm ‘(A −v B)) / 2) → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x) ↔ ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2))))
27	22, 26	imbi12d 474	. . . . . . . 8 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)) ↔ (0 < ((norm ‘(A −v B)) / 2) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)))))
28	27	rcla4v 1402	. . . . . . 7 ⊢ (∀x ∈ ℝ (0 < x → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < x)) → (((norm ‘(A −v B)) / 2) ∈ ℝ → (0 < ((norm ‘(A −v B)) / 2) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)))))
29	19, 21, 28	mp2 43	. . . . . 6 ⊢ (0 < ((norm ‘(A −v B)) / 2) → ∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)))
30	5, 10	hlimconv 5111	. . . . . . . 8 ⊢ (F ⇝v B → ∀x ∈ ℝ (0 < x → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x)))
31	9, 30	ax-mp 6	. . . . . . 7 ⊢ ∀x ∈ ℝ (0 < x → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x))
32		breq2 2066	. . . . . . . . . . . 12 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((norm ‘((F ‘z) −v B)) < x ↔ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)))
33	32	imbi2d 464	. . . . . . . . . . 11 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((w ≤ z → (norm ‘((F ‘z) −v B)) < x) ↔ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
34	33	biraldv 1219	. . . . . . . . . 10 ⊢ (x = ((norm ‘(A −v B)) / 2) → (∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x) ↔ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
35	34	birexdv 1220	. . . . . . . . 9 ⊢ (x = ((norm ‘(A −v B)) / 2) → (∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x) ↔ ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
36	22, 35	imbi12d 474	. . . . . . . 8 ⊢ (x = ((norm ‘(A −v B)) / 2) → ((0 < x → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x)) ↔ (0 < ((norm ‘(A −v B)) / 2) → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)))))
37	36	rcla4v 1402	. . . . . . 7 ⊢ (∀x ∈ ℝ (0 < x → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < x)) → (((norm ‘(A −v B)) / 2) ∈ ℝ → (0 < ((norm ‘(A −v B)) / 2) → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)))))
38	31, 21, 37	mp2 43	. . . . . 6 ⊢ (0 < ((norm ‘(A −v B)) / 2) → ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)))
39	29, 38	jca 236	. . . . 5 ⊢ (0 < ((norm ‘(A −v B)) / 2) → (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
40		r19.26 1289	. . . . . . . . 9 ⊢ (∀z ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) ↔ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
41	14	ltnr 4338	. . . . . . . . . . . . 13 ⊢ ¬ (norm ‘(A −v B)) < (norm ‘(A −v B))
42	5, 6	hlimseq 5109	. . . . . . . . . . . . . . . . . . 19 ⊢ (F ⇝v A → F:ℕ–→ ℋ )
43	4, 42	ax-mp 6	. . . . . . . . . . . . . . . . . 18 ⊢ F:ℕ–→ ℋ
44		ffvrn 2890	. . . . . . . . . . . . . . . . . 18 ⊢ ((F:ℕ–→ ℋ ∧ z ∈ ℕ) → (F ‘z) ∈ ℋ )
45	43, 44	mpan 518	. . . . . . . . . . . . . . . . 17 ⊢ (z ∈ ℕ → (F ‘z) ∈ ℋ )
46		normsubt 5091	. . . . . . . . . . . . . . . . . 18 ⊢ (((F ‘z) ∈ ℋ ∧ A ∈ ℋ ) → (norm ‘((F ‘z) −v A)) = (norm ‘(A −v (F ‘z))))
47	8, 46	mpan2 519	. . . . . . . . . . . . . . . . 17 ⊢ ((F ‘z) ∈ ℋ → (norm ‘((F ‘z) −v A)) = (norm ‘(A −v (F ‘z))))
48	45, 47	syl 12	. . . . . . . . . . . . . . . 16 ⊢ (z ∈ ℕ → (norm ‘((F ‘z) −v A)) = (norm ‘(A −v (F ‘z))))
49	48	breq1d 2071	. . . . . . . . . . . . . . 15 ⊢ (z ∈ ℕ → ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ↔ (norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2)))
50	49	anbi1d 469	. . . . . . . . . . . . . 14 ⊢ (z ∈ ℕ → (((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) ↔ ((norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
51	8, 12	pm3.2i 234	. . . . . . . . . . . . . . . . 17 ⊢ (A ∈ ℋ ∧ B ∈ ℋ )
52		norm3lemt 5097	. . . . . . . . . . . . . . . . 17 ⊢ (((A ∈ ℋ ∧ B ∈ ℋ ) ∧ ((F ‘z) ∈ ℋ ∧ (norm ‘(A −v B)) ∈ ℝ)) → (((norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (norm ‘(A −v B)) < (norm ‘(A −v B))))
53	51, 52	mpan 518	. . . . . . . . . . . . . . . 16 ⊢ (((F ‘z) ∈ ℋ ∧ (norm ‘(A −v B)) ∈ ℝ) → (((norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (norm ‘(A −v B)) < (norm ‘(A −v B))))
54	14, 53	mpan2 519	. . . . . . . . . . . . . . 15 ⊢ ((F ‘z) ∈ ℋ → (((norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (norm ‘(A −v B)) < (norm ‘(A −v B))))
55	45, 54	syl 12	. . . . . . . . . . . . . 14 ⊢ (z ∈ ℕ → (((norm ‘(A −v (F ‘z))) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (norm ‘(A −v B)) < (norm ‘(A −v B))))
56	50, 55	sylbid 178	. . . . . . . . . . . . 13 ⊢ (z ∈ ℕ → (((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (norm ‘(A −v B)) < (norm ‘(A −v B))))
57	41, 56	mtoi 94	. . . . . . . . . . . 12 ⊢ (z ∈ ℕ → ¬ ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)))
58		con3 86	. . . . . . . . . . . . 13 ⊢ (((y ≤ z ∧ w ≤ z) → ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → (¬ ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → ¬ (y ≤ z ∧ w ≤ z)))
59	58	com12 13	. . . . . . . . . . . 12 ⊢ (¬ ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2)) → (((y ≤ z ∧ w ≤ z) → ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ¬ (y ≤ z ∧ w ≤ z)))
60	57, 59	syl 12	. . . . . . . . . . 11 ⊢ (z ∈ ℕ → (((y ≤ z ∧ w ≤ z) → ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ¬ (y ≤ z ∧ w ≤ z)))
61		prth 429	. . . . . . . . . . 11 ⊢ (((y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ((y ≤ z ∧ w ≤ z) → ((norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2) ∧ (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
62	60, 61	syl5 22	. . . . . . . . . 10 ⊢ (z ∈ ℕ → (((y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ¬ (y ≤ z ∧ w ≤ z)))
63	62	r19.20i 1253	. . . . . . . . 9 ⊢ (∀z ∈ ℕ ((y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z))
64	40, 63	sylbir 176	. . . . . . . 8 ⊢ ((∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z))
65	64	r19.22si 1275	. . . . . . 7 ⊢ (∃w ∈ ℕ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z))
66	65	r19.22si 1275	. . . . . 6 ⊢ (∃y ∈ ℕ ∃w ∈ ℕ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ∃y ∈ ℕ ∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z))
67		reeanv 1316	. . . . . 6 ⊢ (∃y ∈ ℕ ∃w ∈ ℕ (∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) ↔ (∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))))
68		ralnex 1209	. . . . . . . . . 10 ⊢ (∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z) ↔ ¬ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
69	68	birex 1224	. . . . . . . . 9 ⊢ (∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z) ↔ ∃w ∈ ℕ ¬ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
70		rexnal 1210	. . . . . . . . 9 ⊢ (∃w ∈ ℕ ¬ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z) ↔ ¬ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
71	69, 70	bitr 151	. . . . . . . 8 ⊢ (∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z) ↔ ¬ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
72	71	birex 1224	. . . . . . 7 ⊢ (∃y ∈ ℕ ∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z) ↔ ∃y ∈ ℕ ¬ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
73		rexnal 1210	. . . . . . 7 ⊢ (∃y ∈ ℕ ¬ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z) ↔ ¬ ∀y ∈ ℕ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
74	72, 73	bitr 151	. . . . . 6 ⊢ (∃y ∈ ℕ ∃w ∈ ℕ ∀z ∈ ℕ ¬ (y ≤ z ∧ w ≤ z) ↔ ¬ ∀y ∈ ℕ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
75	66, 67, 74	3imtr3 191	. . . . 5 ⊢ ((∃y ∈ ℕ ∀z ∈ ℕ (y ≤ z → (norm ‘((F ‘z) −v A)) < ((norm ‘(A −v B)) / 2)) ∧ ∃w ∈ ℕ ∀z ∈ ℕ (w ≤ z → (norm ‘((F ‘z) −v B)) < ((norm ‘(A −v B)) / 2))) → ¬ ∀y ∈ ℕ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
76	17, 39, 75	3syl 21	. . . 4 ⊢ (0 < (norm ‘(A −v B)) → ¬ ∀y ∈ ℕ ∀w ∈ ℕ ∃z ∈ ℕ (y ≤ z ∧ w ≤ z))
77	2, 76	mt2 96	. . 3 ⊢ ¬ 0 < (norm ‘(A −v B))
78		normgt0t 5078	. . . . 5 ⊢ ((A −v B) ∈ ℋ → (¬ (A −v B) = 0v ↔ 0 < (norm ‘(A −v B))))
79	13, 78	ax-mp 6	. . . 4 ⊢ (¬ (A −v B) = 0v ↔ 0 < (norm ‘(A −v B)))
80	79	bicon1i 193	. . 3 ⊢ (¬ 0 < (norm ‘(A −v B)) ↔ (A −v B) = 0v)
81	77, 80	mpbi 164	. 2 ⊢ (A −v B) = 0v
82	8, 12	hvsubeq0 5035	. 2 ⊢ ((A −v B) = 0v ↔ A = B)
83	81, 82	mpbi 164	1 ⊢ A = B

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   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  0_vc0v 4961   −_v cmv 4962  normcno 4964   ⇝_v chli 4966

This theorem is referenced by:  hlimuni 5144

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-hlim 5107

metamath.org