limsuppnflem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsuppnflem		Structured version Visualization version GIF version

Theorem limsuppnflem 44053

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 401	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
3	2	ralbii 3093	. . . . . . . . . . . . 13 ⊢ (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∀𝑗 ∈ 𝐴 ¬ (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
4		ralnex 3072	. . . . . . . . . . . . 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 3094	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ↔ ∃𝑘 ∈ ℝ ¬ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)))
7		rexnal 3100	. . . . . . . . . . 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 3094	. . . . . . . . 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 3100	. . . . . . . . 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 227	. . . . . . 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 1137	. . . . . . . . . . . . 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 408	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → ¬ 𝑥 ≤ (𝐹‘𝑗))
16	15	3adant1 1131	. . . . . . . . . . . . 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 7041	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
19	18	ad4ant14 751	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → (𝐹‘𝑗) ∈ ℝ^*)
20	19	adantr 482	. . . . . . . . . . . . . 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 11211	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℝ^*)
23	21, 22	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → 𝑥 ∈ ℝ^*)
24	23	adantr 482	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
25		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → ¬ 𝑥 ≤ (𝐹‘𝑗))
26	18	ad4ant13 750	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈ ℝ^*)
27	22	ad3antlr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ^*)
28	26, 27	xrltnled 43700	. . . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) < 𝑥)
31	20, 24, 30	xrltled 13080	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ 𝑥)
32	13, 16, 31	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
33	32	3exp 1120	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) → ((𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
34	33	ralimdva 3161	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → ¬ 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
35	34	reximdva 3162	. . . . . . . . 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 3162	. . . . . . . 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 408	. . . . . . 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 597	. . . . . 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 11152	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ V
40	39	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ∈ V)
41		limsuppnflem.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
42	40, 41	ssexd 5287	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ V)
43	17, 42	fexd 7183	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ V)
44	43	limsupcld 44033	. . . . . . . . . 10 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
45	44	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) ∈ ℝ^*)
46	22	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
47		pnfxr 11219	. . . . . . . . . 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 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝐴 ⊆ ℝ)
51	17	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝐹:𝐴⟶ℝ^*)
52		simpr 486	. . . . . . . . . 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 44029	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) ≤ 𝑥)
54		ltpnf 13051	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 𝑥 < +∞)
55	54	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → 𝑥 < +∞)
56	45, 46, 48, 53, 55	xrlelttrd 13090	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) < +∞)
57	56	rexlimdva2 3151	. . . . . . 7 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → (lim sup‘𝐹) < +∞))
58	57	imp 408	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (lim sup‘𝐹) < +∞)
59	38, 58	syldan 592	. . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (lim sup‘𝐹) < +∞)
60	59	adantlr 714	. . . 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 2833	. . . . . . 7 ⊢ ((lim sup‘𝐹) = +∞ → (lim sup‘𝐹) ∈ ℝ^*)
64	63, 61	xreqnltd 43732	. . . . . 6 ⊢ ((lim sup‘𝐹) = +∞ → ¬ (lim sup‘𝐹) < +∞)
65	64	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = +∞) → ¬ (lim sup‘𝐹) < +∞)
66	65	adantr 482	. . . 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 414	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = +∞ → ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))))
69	41	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝐴 ⊆ ℝ)
70	17	adantr 482	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → 𝐹:𝐴⟶ℝ^*)
71		simpr 486	. . . 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 44045	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗))) → (lim sup‘𝐹) = +∞)
73	72	ex 414	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∀𝑘 ∈ ℝ ∃𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 ∧ 𝑥 ≤ (𝐹‘𝑗)) → (lim sup‘𝐹) = +∞))
74	68, 73	impbid 211	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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2883 ∀wral 3061 ∃wrex 3070 Vcvv 3447 ⊆ wss 3914 class class class wbr 5111 ⟶wf 6498 ‘cfv 6502 ℝcr 11060 +∞cpnf 11196 ℝ^*cxr 11198 < clt 11199 ≤ cle 11200 lim supclsp 15365

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2703 ax-rep 5248 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 ax-pre-sup 11139

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-er 8656 df-en 8892 df-dom 8893 df-sdom 8894 df-sup 9388 df-inf 9389 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-ico 13281 df-limsup 15366

This theorem is referenced by: limsuppnf 44054

W3C validator