Theorem limsupmnflem 46169

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 11123	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupmnflem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5262	. . . . 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 46141	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
9	7	rneqi 5887	. . . . . 6 ⊢ ran 𝐺 = ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
10	9	infeq1i 9386	. . . . 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 2772	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ))
13	12	eqeq1d 2739	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
14		nfv 1916	. . 3 ⊢ Ⅎ𝑥𝜑
15	6	fimassd 6684	. . . . 5 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
17	16	supxrcld 45558	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ∈ ℝ*)
18	1, 14, 17	infxrunb3rnmpt 45877	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
19	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
20		ressxr 11183	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ*)
22	21	sselda 3922	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*)
23		supxrleub 13272	. . . . . . 7 ⊢ (((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ∧ 𝑥 ∈ ℝ*) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
24	19, 22, 23	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
25	24	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
26	6	ffnd 6664	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	26	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝐹 Fn 𝐴)
28		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐴)
29	20	sseli 3918	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ*)
30	29	ad3antlr 732	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℝ*)
31		pnfxr 11193	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ)
35	33, 34	sseldd 3923	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ*)
36	35	ad4ant13 752	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℝ*)
37		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
38	34	ltpnfd 13066	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 < +∞)
39	38	ad4ant13 752	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 < +∞)
40	30, 32, 36, 37, 39	elicod 13342	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (𝑘[,)+∞))
41	27, 28, 40	fnfvimad 7183	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
42	41	adantllr 720	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
43		simpllr 776	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
44		breq1 5089	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑗) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
45	44	rspcva 3563	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥)
46	42, 43, 45	syl2anc 585	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
47	46	adantl4r 756	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
48	47	ex 412	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
49	48	ralrimiva 3130	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
50	49	ex 412	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
51		nfcv 2899	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
52	26	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝐹 Fn 𝐴)
53		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ∈ (𝐹 “ (𝑘[,)+∞)))
54	51, 52, 53	fvelimad 6902	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦)
55	54	ad4ant14 753	. . . . . . . . . . . 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 3262	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ (𝑘[,)+∞))
64	60, 61, 63	icogelbd 13344	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗)
65	64	adantlr 716	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 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 592	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
70	69	adantll 715	. . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑗) = 𝑦 → 𝑦 = (𝐹‘𝑗))
74	73	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 = (𝐹‘𝑗))
75		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → (𝐹‘𝑗) ≤ 𝑥)
76	74, 75	eqbrtrd 5108	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
77	76	ex 412	. . . . . . . . . . . . . . . . 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 719	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
80	79	ex 412	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
83	55, 82	syldan 592	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ≤ 𝑥)
84	83	ralrimiva 3130	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
85	84	adantllr 720	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
86	24	ad2antrr 727	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥)
88	87	ex 412	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥))
89	88, 25	sylibd 239	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
90	50, 89	impbid 212	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
91	25, 90	bitrd 279	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
92	91	rexbidva 3160	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
93	92	ralbidva 3159	. 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 307	1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086   ↦ cmpt 5167  ran crn 5626   “ cima 5628   Fn wfn 6488  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  supcsup 9347  infcinf 9348  ℝcr 11031  +∞cpnf 11170  -∞cmnf 11171  ℝ^*cxr 11172   < clt 11173   ≤ cle 11174  [,)cico 13294  lim supclsp 15426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-sup 9349  df-inf 9350  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-ico 13298  df-limsup 15427

This theorem is referenced by:  limsupmnf  46170