limsuppnflem - Mathbox for Glauco Siliprandi

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Theorem limsuppnflem 45818

Description: If the restriction of a function to every upper interval is unbounded above, its lim sup is +∞. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
limsuppnflem.j	⊢ Ⅎ𝑗𝐹
limsuppnflem.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsuppnflem.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)

**Assertion**
Ref	Expression
limsuppnflem	⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))

Distinct variable groups: 𝐴,𝑗,𝑘,𝑥 𝑘,𝐹,𝑥 𝜑,𝑗,𝑘,𝑥 Allowed substitution hint: 𝐹(𝑗)

**Proof of Theorem limsuppnflem**
Step	Hyp	Ref	Expression
1		id 22	. . . . . . 7 ⊢ (𝜑 → 𝜑)
2		imnan 399	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
3	2	ralbii 3078	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
4		ralnex 3058	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
5	3, 4	bitri 275	. . . . . . . . . . . 12 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
6	5	rexbii 3079	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
7		rexnal 3084	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
8	6, 7	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
9	8	rexbii 3079	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
10		rexnal 3084	. . . . . . . . 9 ⊢ (∃𝑥 ∈ ℝ ¬ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
11	9, 10	bitri 275	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
12	11	biimpri 228	. . . . . . 7 ⊢ (¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)))
13		simp1 1136	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴))
14		id 22	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)))
15	14	imp 406	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → ¬ 𝑥 ≤ (𝐹‘𝑗))
16	15	3adant1 1130	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → ¬ 𝑥 ≤ (𝐹‘𝑗))
17		limsuppnflem.f	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
18	17	ffvelcdmda 7017	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
19	18	ad4ant14 752	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
20	19	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
21		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ)
22		rexr 11158	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
24	23	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
25		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → ¬ 𝑥 ≤ (𝐹‘𝑗))
26	18	ad4ant13 751	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
27	22	ad3antlr 731	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
28	26, 27	xrltnled 11180	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → ((𝐹‘𝑗) < 𝑥 ↔ ¬ 𝑥 ≤ (𝐹‘𝑗)))
29	25, 28	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) < 𝑥)
30	29	adantllr 719	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) < 𝑥)
31	20, 24, 30	xrltled 13049	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ 𝑥)
32	13, 16, 31	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
33	32	3exp 1119	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
34	33	ralimdva 3144	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
35	34	reximdva 3145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
36	35	reximdva 3145	. . . . . . . 8 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
37	36	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
38	1, 12, 37	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
39		reex 11097	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
40	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
41		limsuppnflem.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
42	40, 41	ssexd 5260	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
43	17, 42	fexd 7161	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
44	43	limsupcld 45798	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
45	44	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) ∈ ℝ^*)
46	22	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
47		pnfxr 11166	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
48	47	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → +∞ ∈ ℝ^*)
49		limsuppnflem.j	. . . . . . . . . 10 ⊢ Ⅎ𝑗𝐹
50	41	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
51	17	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝐹:𝐴⟶ℝ^*)
52		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
53	49, 50, 51, 46, 52	limsupbnd1f 45794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) ≤ 𝑥)
54		ltpnf 13019	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
55	54	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝑥 < +∞)
56	45, 46, 48, 53, 55	xrlelttrd 13059	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) < +∞)
57	56	rexlimdva2 3135	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (lim sup‘𝐹) < +∞))
58	57	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) < +∞)
59	38, 58	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (lim sup‘𝐹) < +∞)
60	59	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (lim sup‘𝐹) = +∞) ∧ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (lim sup‘𝐹) < +∞)
61		id 22	. . . . . . . 8 ⊢ ((lim sup‘𝐹) = +∞ → (lim sup‘𝐹) = +∞)
62	47	a1i 11	. . . . . . . 8 ⊢ ((lim sup‘𝐹) = +∞ → +∞ ∈ ℝ^*)
63	61, 62	eqeltrd 2831	. . . . . . 7 ⊢ ((lim sup‘𝐹) = +∞ → (lim sup‘𝐹) ∈ ℝ^*)
64	63, 61	xreqnltd 45503	. . . . . 6 ⊢ ((lim sup‘𝐹) = +∞ → ¬ (lim sup‘𝐹) < +∞)
65	64	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = +∞) → ¬ (lim sup‘𝐹) < +∞)
66	65	adantr 480	. . . 4 ⊢ (((𝜑 ∧ (lim sup‘𝐹) = +∞) ∧ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ¬ (lim sup‘𝐹) < +∞)
67	60, 66	condan 817	. . 3 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = +∞) → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
68	67	ex 412	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
69	41	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝐴 ⊆ ℝ)
70	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝐹:𝐴⟶ℝ^*)
71		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
72	49, 69, 70, 71	limsuppnfd 45810	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (lim sup‘𝐹) = +∞)
73	72	ex 412	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (lim sup‘𝐹) = +∞))
74	68, 73	impbid 212	1 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ ↔ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 Ⅎwnfc 2879 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ⊆ wss 3897 class class class wbr 5089 ⟶wf 6477 ‘cfv 6481 ℝcr 11005 +∞cpnf 11143 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 lim supclsp 15377

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

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 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-ico 13251 df-limsup 15378

This theorem is referenced by: limsuppnf 45819

W3C validator