Theorem limsupmnflem 46729

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 1947	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		reex 11291	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupmnflem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5286	. . . . 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 46701	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
9	7	rneqi 5919	. . . . . 6 ⊢ ran 𝐺 = ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
10	9	infeq1i 9471	. . . . 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 2796	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ))
13	12	eqeq1d 2763	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
14		nfv 1947	. . 3 ⊢ Ⅎ𝑥𝜑
15	6	fimassd 6731	. . . . 5 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
16	15	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
17	16	supxrcld 46121	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ∈ ℝ*)
18	1, 14, 17	infxrunb3rnmpt 46437	. 2 ⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
19	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
20		ressxr 11353	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ*)
22	21	sselda 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*)
23		supxrleub 13456	. . . . . . 7 ⊢ (((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ∧ 𝑥 ∈ ℝ*) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
24	19, 22, 23	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
25	24	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
26	6	ffnd 6710	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 Fn 𝐴)
27	26	ad3antrrr 743	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝐹 Fn 𝐴)
28		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐴)
29	20	sseli 3927	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ*)
30	29	ad3antlr 744	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℝ*)
31		pnfxr 11363	. . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ)
35	33, 34	sseldd 3932	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ*)
36	35	ad4ant13 764	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℝ*)
37		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗)
38	34	ltpnfd 13250	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 < +∞)
39	38	ad4ant13 764	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 < +∞)
40	30, 32, 36, 37, 39	elicod 13526	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (𝑘[,)+∞))
41	27, 28, 40	fnfvimad 7240	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
42	41	adantllr 732	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)))
43		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
44		breq1 5106	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑗) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
45	44	rspcva 3575	. . . . . . . . . . 11 ⊢ (((𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥)
46	42, 43, 45	syl2anc 596	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
47	46	adantl4r 768	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥)
48	47	ex 418	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
49	48	ralrimiva 3155	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
50	49	ex 418	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
51		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐹
52	26	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝐹 Fn 𝐴)
53		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ∈ (𝐹 “ (𝑘[,)+∞)))
54	51, 52, 53	fvelimad 6952	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦)
55	54	ad4ant14 765	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦)
56		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ℝ)
57		nfra1 3287	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
58	56, 57	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
59		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗 𝑦 ≤ 𝑥
60	29	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ∈ ℝ*)
61	31	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → +∞ ∈ ℝ*)
62		elinel2 4148	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ (𝑘[,)+∞))
63	62	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ (𝑘[,)+∞))
64	60, 61, 63	icogelbd 13528	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗)
65	64	adantlr 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗)
66		elinel1 4147	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ 𝐴)
67	66	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ 𝐴)
68		rspa 3252	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
69	67, 68	syldan 603	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
70	69	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
71	65, 70	mpd 16	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑗) ≤ 𝑥)
72		id 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹‘𝑗) = 𝑦 → (𝐹‘𝑗) = 𝑦)
73	72	eqcomd 2767	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑗) = 𝑦 → 𝑦 = (𝐹‘𝑗))
74	73	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 = (𝐹‘𝑗))
75		simpl 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → (𝐹‘𝑗) ≤ 𝑥)
76	74, 75	eqbrtrd 5127	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
77	76	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑗) ≤ 𝑥 → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
78	71, 77	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ℝ ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
79	78	adantlll 731	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
80	79	ex 418	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)))
81	58, 59, 80	rexlimd 3270	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))
82	81	imp 412	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥)
83	55, 82	syldan 603	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ≤ 𝑥)
84	83	ralrimiva 3155	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
85	84	adantllr 732	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)
86	24	ad2antrr 739	. . . . . . . . 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 260	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥)
88	87	ex 418	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥))
89	88, 25	sylibd 242	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥))
90	50, 89	impbid 215	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
91	25, 90	bitrd 282	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
92	91	rexbidva 3185	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
93	92	ralbidva 3184	. 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 310	1 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ran crn 5652   “ cima 5654   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  supcsup 9432  infcinf 9433  ℝcr 11199  +∞cpnf 11340  -∞cmnf 11341  ℝ^*cxr 11342   < clt 11343   ≤ cle 11344  [,)cico 13478  lim supclsp 15637

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-ico 13482  df-limsup 15638

This theorem is used by:  limsupmnf  46730