Theorem limsupmnflem 43597

Description: The superior limit of a function is -∞ if and only if every real number is the upper bound of the restriction of the function to an upper interval of real numbers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
limsupmnflem.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
limsupmnflem.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
limsupmnflem.g	⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))

Assertion

Ref	Expression
limsupmnflem	⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

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

Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐺(𝑥,𝑗,𝑘)

Proof of Theorem limsupmnflem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		reex 11063	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupmnflem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5268	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
6		limsupmnflem.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*)
7		limsupmnflem.g	. . . . 5 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
8	1, 5, 6, 7	limsupval3 43569	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
9	7	rneqi 5878	. . . . . 6 ⊢ ran 𝐺 = ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
10	9	infeq1i 9335	. . . . 5 ⊢ inf(ran 𝐺, ℝ*, < ) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < )
11	10	a1i 11	. . . 4 ⊢ (𝜑 → inf(ran 𝐺, ℝ*, < ) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ))
12	8, 11	eqtrd 2776	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ))
13	12	eqeq1d 2738	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
14		nfv 1916	. . 3 ⊢ Ⅎ𝑥𝜑
15	6	fimassd 43099	. . . . 5 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
17	16	supxrcld 42977	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ∈ ℝ*)
18	1, 14, 17	infxrunb3rnmpt 43303	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
19	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
20		ressxr 11120	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ*)
22	21	sselda 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*)
23		supxrleub 13161	. . . . . . 7 ⊢ (((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ∧ 𝑥 ∈ ℝ*) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
24	19, 22, 23	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
25	24	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
26	6	ffnd 6652	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	26	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝐹 Fn 𝐴)
28		simplr 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐴)
29	20	sseli 3928	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ*)
30	29	ad3antlr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℝ*)
31		pnfxr 11130	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ*
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → +∞ ∈ ℝ*)
33	20	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ℝ ⊆ ℝ*)
34	4	sselda 3932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ)
35	33, 34	sseldd 3933	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ*)
36	35	ad4ant13 748	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℝ*)
37		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
38	34	ltpnfd 12958	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 < +∞)
39	38	ad4ant13 748	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 < +∞)
40	30, 32, 36, 37, 39	elicod 13230	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (𝑘[,)+∞))
41	27, 28, 40	fnfvimad 7166	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
42	41	adantllr 716	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
43		simpllr 773	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
44		breq1 5095	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑗) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
45	44	rspcva 3568	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥)
46	42, 43, 45	syl2anc 584	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
47	46	adantl4r 752	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
48	47	ex 413	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
49	48	ralrimiva 3139	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
50	49	ex 413	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
51		nfcv 2904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
52	26	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝐹 Fn 𝐴)
53		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ∈ (𝐹 “ (𝑘[,)+∞)))
54	51, 52, 53	fvelimad 6892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦)
55	54	ad4ant14 749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦)
56		nfv 1916	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ℝ)
57		nfra1 3263	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
58	56, 57	nfan 1901	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
59		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑦 ≤ 𝑥
60	29	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ∈ ℝ*)
61	31	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → +∞ ∈ ℝ*)
62		elinel2 4143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ (𝑘[,)+∞))
63	62	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ (𝑘[,)+∞))
64	60, 61, 63	icogelbd 43432	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗)
65	64	adantlr 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗)
66		elinel1 4142	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ 𝐴)
67	66	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ 𝐴)
68		rspa 3227	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
69	67, 68	syldan 591	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
70	69	adantll 711	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
71	65, 70	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑗) ≤ 𝑥)
72		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑗) = 𝑦 → (𝐹‘𝑗) = 𝑦)
73	72	eqcomd 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑗) = 𝑦 → 𝑦 = (𝐹‘𝑗))
74	73	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 = (𝐹‘𝑗))
75		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → (𝐹‘𝑗) ≤ 𝑥)
76	74, 75	eqbrtrd 5114	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
77	76	ex 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑗) ≤ 𝑥 → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
78	71, 77	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
79	78	adantlll 715	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
80	79	ex 413	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)))
81	58, 59, 80	rexlimd 3245	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
82	81	imp 407	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
83	55, 82	syldan 591	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ≤ 𝑥)
84	83	ralrimiva 3139	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
85	84	adantllr 716	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
86	24	ad2antrr 723	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
87	85, 86	mpbird 256	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥)
88	87	ex 413	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥))
89	88, 25	sylibd 238	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
90	50, 89	impbid 211	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
91	25, 90	bitrd 278	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
92	91	rexbidva 3169	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
93	92	ralbidva 3168	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
94	13, 18, 93	3bitr2d 306	1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∃wrex 3070  Vcvv 3441   ∩ cin 3897   ⊆ wss 3898   class class class wbr 5092   ↦ cmpt 5175  ran crn 5621   “ cima 5623   Fn wfn 6474  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337  supcsup 9297  infcinf 9298  ℝcr 10971  +∞cpnf 11107  -∞cmnf 11108  ℝ^*cxr 11109   < clt 11110   ≤ cle 11111  [,)cico 13182  lim supclsp 15278

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-ico 13186  df-limsup 15279

This theorem is referenced by:  limsupmnf  43598