Theorem limsupreuz 43278

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 2907	. . . 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 42924	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
6	1, 2, 3, 5	limsupre3uzlem 43276	. . 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 5077	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
8	7	rexbidv 3226	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
9	8	ralbidv 3112	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
10		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
11	10	rexeqdv 3349	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙)))
12		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑥
13		nfcv 2907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 ≤
14		limsupreuz.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
15		nfcv 2907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑙
16	14, 15	nffv 6784	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑙)
17	12, 13, 16	nfbr 5121	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
18		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗)
19		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
20	19	breq2d 5086	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
21	17, 18, 20	cbvrexw 3374	. . . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
24	23	cbvralvw 3383	. . . . . . . 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 278	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
27	26	cbvrexvw 3384	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
28		breq2 5078	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
29	28	ralbidv 3112	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
30	29	rexbidv 3226	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
31	10	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥))
32	16, 13, 12	nfbr 5121	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
33		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
34	19	breq1d 5084	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
35	32, 33, 34	cbvralw 3373	. . . . . . . . . . 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 278	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
38	37	cbvrexvw 3384	. . . . . . . 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 278	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
41	40	cbvrexvw 3384	. . . . 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 627	. . . 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 278	. 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 1917	. . . . . . . 8 ⊢ Ⅎ𝑖(𝐹‘𝑗) ≤ 𝑥
46		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑖
47	14, 46	nffv 6784	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑖)
48	47, 13, 12	nfbr 5121	. . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑖) ≤ 𝑥
49		fveq2 6774	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
50	49	breq1d 5084	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥))
51	45, 48, 50	cbvralw 3373	. . . . . . 7 ⊢ (∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
52	51	rexbii 3181	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
53	52	rexbii 3181	. . . . 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 1917	. . . . 5 ⊢ Ⅎ𝑖𝜑
56	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
57		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
58	56, 57	ffvelrnd 6962	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ ℝ)
59	55, 2, 3, 58	uzub 42971	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥))
60		eqcom 2745	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 ↔ 𝑖 = 𝑗)
61	60	imbi1i 350	. . . . . . . . 9 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)))
62		bicom 221	. . . . . . . . . 10 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥) ↔ ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
63	62	imbi2i 336	. . . . . . . . 9 ⊢ ((𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
64	61, 63	bitri 274	. . . . . . . 8 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
65	50, 64	mpbi 229	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
66	48, 45, 65	cbvralw 3373	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
67	66	rexbii 3181	. . . . 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 305	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))
70	69	anbi2d 629	. 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 278	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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  ∃wrex 3065   class class class wbr 5074  ⟶wf 6429  ‘cfv 6433  ℝcr 10870   ≤ cle 11010  ℤcz 12319  ℤ_≥cuz 12582  lim supclsp 15179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-ico 13085  df-fz 13240  df-fzo 13383  df-fl 13512  df-ceil 13513  df-limsup 15180

This theorem is referenced by:  limsupreuzmpt  43280  limsupgtlem  43318