Theorem liminfreuzlem 46230

Description: Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
liminfreuzlem.1	⊢ Ⅎ𝑗𝐹
liminfreuzlem.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
liminfreuzlem.3	⊢ 𝑍 = (ℤ≥‘𝑀)
liminfreuzlem.4	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)

Assertion

Ref	Expression
liminfreuzlem	⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))

Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘,𝑥   𝜑,𝑗,𝑘,𝑥

Allowed substitution hints:   𝐹(𝑗)   𝑀(𝑥,𝑘)

Proof of Theorem liminfreuzlem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1916	. . . . 5 ⊢ Ⅎ𝑗𝜑
2		liminfreuzlem.1	. . . . 5 ⊢ Ⅎ𝑗𝐹
3		liminfreuzlem.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		liminfreuzlem.3	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		liminfreuzlem.4	. . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
6	1, 2, 3, 4, 5	liminfvaluz4 46227	. . . 4 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))))
7	6	eleq1d 2821	. . 3 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
8	4	fvexi 6854	. . . . . . 7 ⊢ 𝑍 ∈ V
9	8	mptex 7178	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V
10		limsupcl 15435	. . . . . 6 ⊢ ((𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V → (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ*)
11	9, 10	ax-mp 5	. . . . 5 ⊢ (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ*
12	11	a1i 11	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ*)
13	12	xnegred 45898	. . 3 ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
14	7, 13	bitr4d 282	. 2 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
15	5	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
16	15	renegcld 11577	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → -(𝐹‘𝑗) ∈ ℝ)
17	1, 3, 4, 16	limsupreuzmpt 46167	. . 3 ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)))
18		renegcl 11457	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
19	18	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → -𝑦 ∈ ℝ)
20		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑦 ∈ ℝ)
21	5	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶ℝ)
22	4	uztrn2 12807	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
23	22	adantll 715	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
24	21, 23	ffvelcdmd 7037	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
25	24	adantllr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
26	20, 25	leneg2d 45876	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑦 ≤ -(𝐹‘𝑗) ↔ (𝐹‘𝑗) ≤ -𝑦))
27	26	rexbidva 3159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
28	27	ralbidva 3158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
29	28	biimpd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
30	29	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦)
31		breq2 5089	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ -𝑦))
32	31	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
33	32	ralbidv 3160	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
34	33	rspcev 3564	. . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
35	19, 30, 34	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)
36	35	rexlimdva2 3140	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37		renegcl 11457	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
38	37	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
39	24	adantllr 720	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
40		simpllr 776	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
41	39, 40	lenegd 11729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹‘𝑗)))
42	41	rexbidva 3159	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
43	42	ralbidva 3158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
44	43	biimpd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
45	44	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗))
46		breq1 5088	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹‘𝑗) ↔ -𝑥 ≤ -(𝐹‘𝑗)))
47	46	rexbidv 3161	. . . . . . . . 9 ⊢ (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
48	47	ralbidv 3160	. . . . . . . 8 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
49	48	rspcev 3564	. . . . . . 7 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗))
50	38, 45, 49	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗))
51	50	rexlimdva2 3140	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)))
52	36, 51	impbid 212	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
53	18	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
54	15	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
55		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑦 ∈ ℝ)
56	54, 55	leneg3d 45885	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (-(𝐹‘𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹‘𝑗)))
57	56	ralbidva 3158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
58	57	biimpd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
59	58	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗))
60		breq1 5088	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ -𝑦 ≤ (𝐹‘𝑗)))
61	60	ralbidv 3160	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
62	61	rspcev 3564	. . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
63	53, 59, 62	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
64	63	rexlimdva2 3140	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
65	37	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → -𝑥 ∈ ℝ)
66		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
67	15	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67	lenegd 11729	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 ≤ (𝐹‘𝑗) ↔ -(𝐹‘𝑗) ≤ -𝑥))
69	68	ralbidva 3158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
70	69	biimpd 229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
71	70	imp 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥)
72		brralrspcev 5145	. . . . . . 7 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
73	65, 71, 72	syl2anc 585	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
74	73	rexlimdva2 3140	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦))
75	64, 74	impbid 212	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
76	52, 75	anbi12d 633	. . 3 ⊢ (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
77	17, 76	bitrd 279	. 2 ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))
78	14, 77	bitrd 279	1 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3061  Vcvv 3429   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  ℝcr 11037  ℝ^*cxr 11178   ≤ cle 11180  -cneg 11378  ℤcz 12524  ℤ_≥cuz 12788  -_𝑒cxne 13060  lim supclsp 15432  lim infclsi 46179

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-q 12899  df-xneg 13063  df-ico 13304  df-fz 13462  df-fzo 13609  df-fl 13751  df-ceil 13752  df-limsup 15433  df-liminf 46180

This theorem is referenced by:  liminfreuz  46231