Theorem limsupreuz 46471

Description: Given a function on the reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is less than or equal to the function, infinitely often; 2. there is a real number that is greater than or equal to the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupreuz.1	⊢ Ⅎ𝑗𝐹
limsupreuz.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
limsupreuz.3	⊢ 𝑍 = (ℤ≥‘𝑀)
limsupreuz.4	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)

Assertion

Ref	Expression
limsupreuz	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)))

Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑍,𝑘,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)

Proof of Theorem limsupreuz

Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2925	. . . 4 ⊢ Ⅎ𝑙𝐹
2		limsupreuz.2	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		limsupreuz.3	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
4		limsupreuz.4	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
5	4	frexr 46120	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
6	1, 2, 3, 5	limsupre3uzlem 46469	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦)))
7		breq1 5112	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
8	7	rexbidv 3189	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
9	8	ralbidv 3188	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
10		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
11	10	rexeqdv 3324	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙)))
12		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑥
13		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 ≤
14		limsupreuz.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
15		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑙
16	14, 15	nffv 6891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑙)
17	12, 13, 16	nfbr 5158	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
18		nfv 1944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗)
19		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
20	19	breq2d 5121	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
21	17, 18, 20	cbvrexw 3308	. . . . . . . . . . 11 ⊢ (∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
23	11, 22	bitrd 282	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
24	23	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
25	24	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
26	9, 25	bitrd 282	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
27	26	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
28		breq2 5113	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
29	28	ralbidv 3188	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
30	29	rexbidv 3189	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
31	10	raleqdv 3323	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥))
32	16, 13, 12	nfbr 5158	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
33		nfv 1944	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
34	19	breq1d 5119	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
35	32, 33, 34	cbvralw 3307	. . . . . . . . . . 11 ⊢ (∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
36	35	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37	31, 36	bitrd 282	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
38	37	cbvrexvw 3244	. . . . . . . 8 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
39	38	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
40	30, 39	bitrd 282	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
41	40	cbvrexvw 3244	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
42	27, 41	anbi12i 639	. . . 4 ⊢ ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
43	42	a1i 11	. . 3 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))
44	6, 43	bitrd 282	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)))
45		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑖(𝐹‘𝑗) ≤ 𝑥
46		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑖
47	14, 46	nffv 6891	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑖)
48	47, 13, 12	nfbr 5158	. . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑖) ≤ 𝑥
49		fveq2 6881	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
50	49	breq1d 5119	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥))
51	45, 48, 50	cbvralw 3307	. . . . . . 7 ⊢ (∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
52	51	rexbii 3112	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
53	52	rexbii 3112	. . . . 5 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
54	53	a1i 11	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥))
55		nfv 1944	. . . . 5 ⊢ Ⅎ𝑖𝜑
56	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
57		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
58	56, 57	ffvelcdmd 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ ℝ)
59	55, 2, 3, 58	uzub 46165	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥))
60		eqcom 2770	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 ↔ 𝑖 = 𝑗)
61	60	imbi1i 352	. . . . . . . . 9 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)))
62		bicom 225	. . . . . . . . . 10 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥) ↔ ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
63	62	imbi2i 339	. . . . . . . . 9 ⊢ ((𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
64	61, 63	bitri 278	. . . . . . . 8 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
65	50, 64	mpbi 233	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
66	48, 45, 65	cbvralw 3307	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
67	66	rexbii 3112	. . . . 5 ⊢ (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
68	67	a1i 11	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))
69	54, 59, 68	3bitrd 308	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))
70	69	anbi2d 641	. 2 ⊢ (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)))
71	44, 70	bitrd 282	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079  ∃wrex 3089   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  ℝcr 11094   ≤ cle 11239  ℤcz 12586  ℤ_≥cuz 12857  lim supclsp 15517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-ico 13373  df-fz 13531  df-fzo 13679  df-fl 13821  df-ceil 13822  df-limsup 15518

This theorem is referenced by:  limsupreuzmpt  46473  limsupgtlem  46511