Theorem limsupre2lem 45762

Description: Given a function on the extended reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller than the function, at some point, in any upper part of the reals; 2. there is a real number that is eventually larger than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupre2lem.1	⊢ Ⅎ𝑗𝐹
limsupre2lem.2	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupre2lem.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)

Assertion

Ref	Expression
limsupre2lem	⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))

Distinct variable groups:   𝐴,𝑗,𝑘,𝑥   𝑘,𝐹,𝑥   𝜑,𝑗,𝑘,𝑥

Allowed substitution hint:   𝐹(𝑗)

Proof of Theorem limsupre2lem

Step	Hyp	Ref	Expression
1		limsupre2lem.3	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
2		reex 11092	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupre2lem.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5257	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
6	1, 5	fexd 7156	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
7	6	limsupcld 45728	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ*)
8		xrre4 45449	. . 3 ⊢ ((lim sup‘𝐹) ∈ ℝ* → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
9	7, 8	syl 17	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ ((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞)))
10		df-ne 2929	. . . . 5 ⊢ ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞)
11	10	a1i 11	. . . 4 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ¬ (lim sup‘𝐹) = -∞))
12		limsupre2lem.1	. . . . . 6 ⊢ Ⅎ𝑗𝐹
13	12, 4, 1	limsupmnf 45759	. . . . 5 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
14	13	notbid 318	. . . 4 ⊢ (𝜑 → (¬ (lim sup‘𝐹) = -∞ ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
15		annim 403	. . . . . . . . . . . 12 ⊢ ((𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
16	15	rexbii 3079	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
17		rexnal 3084	. . . . . . . . . . 11 ⊢ (∃𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
18	16, 17	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
19	18	ralbii 3078	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
20		ralnex 3058	. . . . . . . . 9 ⊢ (∀𝑘 ∈ ℝ ¬ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
21	19, 20	bitri 275	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
22	21	rexbii 3079	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
23		rexnal 3084	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ¬ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
24	22, 23	bitr2i 276	. . . . . 6 ⊢ (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥))
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥)))
26		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ)
27	26	rexrd 11157	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ*)
28	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝐹:𝐴⟶ℝ*)
29	28	ffvelcdmda 7012	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ*)
30	27, 29	xrltnled 11175	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝑥 < (𝐹‘𝑗) ↔ ¬ (𝐹‘𝑗) ≤ 𝑥))
31	30	bicomd 223	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (¬ (𝐹‘𝑗) ≤ 𝑥 ↔ 𝑥 < (𝐹‘𝑗)))
32	31	anbi2d 630	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
33	32	rexbidva 3154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
34	33	ralbidv 3155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
35	34	rexbidva 3154	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ ¬ (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
36	25, 35	bitrd 279	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
37	11, 14, 36	3bitrd 305	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ -∞ ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗))))
38		df-ne 2929	. . . . 5 ⊢ ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞)
39	38	a1i 11	. . . 4 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ¬ (lim sup‘𝐹) = +∞))
40	12, 4, 1	limsuppnf 45749	. . . . 5 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
41	40	notbid 318	. . . 4 ⊢ (𝜑 → (¬ (lim sup‘𝐹) = +∞ ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
42	29, 27	xrltnled 11175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝐹‘𝑗) < 𝑥 ↔ ¬ 𝑥 ≤ (𝐹‘𝑗)))
43	42	imbi2d 340	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
44	43	ralbidva 3153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
45	44	rexbidv 3156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
46	45	rexbidva 3154	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))))
47		imnan 399	. . . . . . . . . . . 12 ⊢ ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
48	47	ralbii 3078	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
49		ralnex 3058	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
50	48, 49	bitri 275	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
51	50	rexbii 3079	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
52		rexnal 3084	. . . . . . . . 9 ⊢ (∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
53	51, 52	bitri 275	. . . . . . . 8 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
54	53	rexbii 3079	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
55		rexnal 3084	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
56	54, 55	bitri 275	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
57	56	a1i 11	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
58	46, 57	bitr2d 280	. . . 4 ⊢ (𝜑 → (¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
59	39, 41, 58	3bitrd 305	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ≠ +∞ ↔ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥)))
60	37, 59	anbi12d 632	. 2 ⊢ (𝜑 → (((lim sup‘𝐹) ≠ -∞ ∧ (lim sup‘𝐹) ≠ +∞) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))
61	9, 60	bitrd 279	1 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 < (𝐹‘𝑗)) ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) < 𝑥))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897   class class class wbr 5086  ⟶wf 6472  ‘cfv 6476  ℝcr 11000  +∞cpnf 11138  -∞cmnf 11139  ℝ^*cxr 11140   < clt 11141   ≤ cle 11142  lim supclsp 15372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078  ax-pre-sup 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-po 5519  df-so 5520  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-sup 9321  df-inf 9322  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-ico 13246  df-limsup 15373

This theorem is referenced by:  limsupre2  45763