Theorem limsupmnflem 47003

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 1949	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		reex 11330	. . . . . . 7 ⊢ ℝ ∈ V
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ V)
4		limsupmnflem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
5	3, 4	ssexd 5293	. . . . 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 46975	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, < ))
9	7	rneqi 5931	. . . . . 6 ⊢ ran 𝐺 = ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))
10	9	infeq1i 9508	. . . . 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 2798	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ))
13	12	eqeq1d 2765	. 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )), ℝ*, < ) = -∞))
14		nfv 1949	. . 3 ⊢ Ⅎ𝑥𝜑
15	6	fimassd 6761	. . . . 5 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
16	15	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*)
17	16	supxrcld 46395	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ∈ ℝ*)
18	1, 14, 17	infxrunb3rnmpt 46711	. 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 11392	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ*
21	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℝ*)
22	21	sselda 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*)
23		supxrleub 13496	. . . . . . 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 6740	. . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℝ → 𝑘 ∈ ℝ*)
30	29	ad3antlr 744	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℝ*)
31		pnfxr 11402	. . . . . . . . . . . . . . 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 3913	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ)
35	33, 34	sseldd 3914	. . . . . . . . . . . . . . 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 13290	. . . . . . . . . . . . . . 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 13566	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (𝑘[,)+∞))
41	27, 28, 40	fnfvimad 7271	. . . . . . . . . . . 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 5111	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑗) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥))
45	44	rspcva 3578	. . . . . . . . . . 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 3158	. . . . . . 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 2926	. . . . . . . . . . . . . 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 6983	. . . . . . . . . . . . 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 1949	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ℝ)
57		nfra1 3290	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑗∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)
58	56, 57	nfan 1934	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))
59		nfv 1949	. . . . . . . . . . . . . 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 4102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ (𝑘[,)+∞))
63	62	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ (𝑘[,)+∞))
64	60, 61, 63	icogelbd 13568	. . . . . . . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . . . . . . . . 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 3255	. . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . . . . . . 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 3273	. . . . . . . . . . . . 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 3158	. . . . . . . . . 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 3188	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) ≤ 𝑥 ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)))
93	92	ralbidva 3187	. 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 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5108   ↦ cmpt 5192  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  supcsup 9469  infcinf 9470  ℝcr 11238  +∞cpnf 11379  -∞cmnf 11380  ℝ^*cxr 11381   < clt 11382   ≤ cle 11383  [,)cico 13518  lim supclsp 15681

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316  ax-pre-sup 11317

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-po 5566  df-so 5567  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-sup 9471  df-inf 9472  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-ico 13522  df-limsup 15682

This theorem is used by:  limsupmnf  47004