Theorem limsupreuz 41455

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 2932	. . . 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 41091	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
6	1, 2, 3, 5	limsupre3uzlem 41453	. . 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 4932	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑙)))
8	7	rexbidv 3242	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
9	8	ralbidv 3147	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙)))
10		fveq2 6499	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
11	10	rexeqdv 3356	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑙 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑙)))
12		nfcv 2932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝑥
13		nfcv 2932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 ≤
14		limsupreuz.1	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
15		nfcv 2932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝑙
16	14, 15	nffv 6509	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗(𝐹‘𝑙)
17	12, 13, 16	nfbr 4976	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗 𝑥 ≤ (𝐹‘𝑙)
18		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙 𝑥 ≤ (𝐹‘𝑗)
19		fveq2 6499	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑗 → (𝐹‘𝑙) = (𝐹‘𝑗))
20	19	breq2d 4941	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → (𝑥 ≤ (𝐹‘𝑙) ↔ 𝑥 ≤ (𝐹‘𝑗)))
21	17, 18, 20	cbvrex 3380	. . . . . . . . . . 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 271	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑙) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
24	23	cbvralv 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 271	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗)))
27	26	cbvrexv 3384	. . . . 5 ⊢ (∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑙 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑥 ≤ (𝐹‘𝑗))
28		breq2 4933	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙) ≤ 𝑦 ↔ (𝐹‘𝑙) ≤ 𝑥))
29	28	ralbidv 3147	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
30	29	rexbidv 3242	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥))
31	10	raleqdv 3355	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ≥‘𝑘)(𝐹‘𝑙) ≤ 𝑥))
32	16, 13, 12	nfbr 4976	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗(𝐹‘𝑙) ≤ 𝑥
33		nfv 1873	. . . . . . . . . . . 12 ⊢ Ⅎ𝑙(𝐹‘𝑗) ≤ 𝑥
34	19	breq1d 4939	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑗 → ((𝐹‘𝑙) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
35	32, 33, 34	cbvral 3379	. . . . . . . . . . 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 271	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
38	37	cbvrexv 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 271	. . . . . 6 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑙 ∈ (ℤ≥‘𝑖)(𝐹‘𝑙) ≤ 𝑦 ↔ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
41	40	cbvrexv 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 617	. . . 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 271	. 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 1873	. . . . . . . 8 ⊢ Ⅎ𝑖(𝐹‘𝑗) ≤ 𝑥
46		nfcv 2932	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝑖
47	14, 46	nffv 6509	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝐹‘𝑖)
48	47, 13, 12	nfbr 4976	. . . . . . . 8 ⊢ Ⅎ𝑗(𝐹‘𝑖) ≤ 𝑥
49		fveq2 6499	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝐹‘𝑗) = (𝐹‘𝑖))
50	49	breq1d 4939	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥))
51	45, 48, 50	cbvral 3379	. . . . . . 7 ⊢ (∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
52	51	rexbii 3194	. . . . . 6 ⊢ (∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥)
53	52	rexbii 3194	. . . . 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 1873	. . . . 5 ⊢ Ⅎ𝑖𝜑
56	4	adantr 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹:𝑍⟶ℝ)
57		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
58	56, 57	ffvelrnd 6677	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖) ∈ ℝ)
59	55, 2, 3, 58	uzub 41142	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑖 ∈ (ℤ≥‘𝑘)(𝐹‘𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥))
60		eqcom 2785	. . . . . . . . . 10 ⊢ (𝑗 = 𝑖 ↔ 𝑖 = 𝑗)
61	60	imbi1i 342	. . . . . . . . 9 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)))
62		bicom 214	. . . . . . . . . 10 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥) ↔ ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
63	62	imbi2i 328	. . . . . . . . 9 ⊢ ((𝑖 = 𝑗 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
64	61, 63	bitri 267	. . . . . . . 8 ⊢ ((𝑗 = 𝑖 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)))
65	50, 64	mpbi 222	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝐹‘𝑖) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
66	48, 45, 65	cbvral 3379	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑍 (𝐹‘𝑖) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥)
67	66	rexbii 3194	. . . . 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 297	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 (𝐹‘𝑗) ≤ 𝑥))
70	69	anbi2d 619	. 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 271	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 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Ⅎwnfc 2916  ∀wral 3088  ∃wrex 3089   class class class wbr 4929  ⟶wf 6184  ‘cfv 6188  ℝcr 10334   ≤ cle 10475  ℤcz 11793  ℤ_≥cuz 12058  lim supclsp 14688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-oadd 7909  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-sup 8701  df-inf 8702  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-nn 11440  df-n0 11708  df-z 11794  df-uz 12059  df-ico 12560  df-fz 12709  df-fzo 12850  df-fl 12977  df-ceil 12978  df-limsup 14689

This theorem is referenced by:  limsupreuzmpt  41457  limsupgtlem  41495