limsuppnflem - Mathbox for Glauco Siliprandi

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Theorem limsuppnflem 45706

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 3083	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
4		ralnex 3063	. . . . . . . . . . . . 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 3084	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
7		rexnal 3090	. . . . . . . . . . 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 3084	. . . . . . . . 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 3090	. . . . . . . . 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 7079	. . . . . . . . . . . . . . . 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 11286	. . . . . . . . . . . . . . . 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 45357	. . . . . . . . . . . . . . . 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 13171	. . . . . . . . . . . . 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 3153	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
35	34	reximdva 3154	. . . . . . . . 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 3154	. . . . . . . 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 11225	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
40	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
41		limsuppnflem.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
42	40, 41	ssexd 5299	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
43	17, 42	fexd 7224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
44	43	limsupcld 45686	. . . . . . . . . 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 11294	. . . . . . . . . 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 45682	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) ≤ 𝑥)
54		ltpnf 13141	. . . . . . . . . 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 13181	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) < +∞)
57	56	rexlimdva2 3144	. . . . . . 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 2835	. . . . . . 7 ⊢ ((lim sup‘𝐹) = +∞ → (lim sup‘𝐹) ∈ ℝ^*)
64	63, 61	xreqnltd 45389	. . . . . 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 45698	. . 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 1540 ∈ wcel 2109 Ⅎwnfc 2884 ∀wral 3052 ∃wrex 3061 Vcvv 3464 ⊆ wss 3931 class class class wbr 5124 ⟶wf 6532 ‘cfv 6536 ℝcr 11133 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 lim supclsp 15491

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 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-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-ico 13373 df-limsup 15492

This theorem is referenced by: limsuppnf 45707

W3C validator