Theorem liminfreuzlem 46259

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 1922	. . . . 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 46256	. . . 4 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))))
7	6	eleq1d 2826	. . 3 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
8	4	fvexi 6845	. . . . . . 7 ⊢ 𝑍 ∈ V
9	8	mptex 7171	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V
10		limsupcl 15430	. . . . . 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 45927	. . 3 ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
14	7, 13	bitr4d 284	. 2 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
15	5	ffvelcdmda 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
16	15	renegcld 11572	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → -(𝐹‘𝑗) ∈ ℝ)
17	1, 3, 4, 16	limsupreuzmpt 46196	. . 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 11452	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
19	18	ad2antlr 734	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → -𝑦 ∈ ℝ)
20		simpllr 782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑦 ∈ ℝ)
21	5	ad2antrr 733	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶ℝ)
22	4	uztrn2 12802	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
23	22	adantll 721	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
24	21, 23	ffvelcdmd 7030	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
25	24	adantllr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
26	20, 25	leneg2d 45905	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑦 ≤ -(𝐹‘𝑗) ↔ (𝐹‘𝑗) ≤ -𝑦))
27	26	rexbidva 3163	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
28	27	ralbidva 3162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
29	28	biimpd 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
30	29	imp 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦)
31		breq2 5079	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ -𝑦))
32	31	rexbidv 3165	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
33	32	ralbidv 3164	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
34	33	rspcev 3562	. . . . . . 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 591	. . . . . 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 3144	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37		renegcl 11452	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
38	37	ad2antlr 734	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
39	24	adantllr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
40		simpllr 782	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
41	39, 40	lenegd 11724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹‘𝑗)))
42	41	rexbidva 3163	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
43	42	ralbidva 3162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
44	43	biimpd 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
45	44	imp 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗))
46		breq1 5078	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹‘𝑗) ↔ -𝑥 ≤ -(𝐹‘𝑗)))
47	46	rexbidv 3165	. . . . . . . . 9 ⊢ (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
48	47	ralbidv 3164	. . . . . . . 8 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
49	48	rspcev 3562	. . . . . . 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 591	. . . . . 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 3144	. . . . 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 214	. . . 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 734	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
54	15	adantlr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
55		simplr 775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑦 ∈ ℝ)
56	54, 55	leneg3d 45914	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (-(𝐹‘𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹‘𝑗)))
57	56	ralbidva 3162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
58	57	biimpd 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
59	58	imp 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗))
60		breq1 5078	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ -𝑦 ≤ (𝐹‘𝑗)))
61	60	ralbidv 3164	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
62	61	rspcev 3562	. . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
63	53, 59, 62	syl2anc 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
64	63	rexlimdva2 3144	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
65	37	ad2antlr 734	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → -𝑥 ∈ ℝ)
66		simplr 775	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
67	15	adantlr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67	lenegd 11724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 ≤ (𝐹‘𝑗) ↔ -(𝐹‘𝑗) ≤ -𝑥))
69	68	ralbidva 3162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
70	69	biimpd 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
71	70	imp 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥)
72		brralrspcev 5135	. . . . . . 7 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
73	65, 71, 72	syl2anc 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
74	73	rexlimdva2 3144	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦))
75	64, 74	impbid 214	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
76	52, 75	anbi12d 639	. . 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 281	. 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 281	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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Ⅎwnfc 2888  ∀wral 3055  ∃wrex 3065  Vcvv 3433   class class class wbr 5075   ↦ cmpt 5156  ⟶wf 6485  ‘cfv 6489  ℝcr 11032  ℝ^*cxr 11173   ≤ cle 11175  -cneg 11373  ℤcz 12519  ℤ_≥cuz 12783  -_𝑒cxne 13055  lim supclsp 15427  lim infclsi 46208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-n0 12433  df-z 12520  df-uz 12784  df-q 12894  df-xneg 13058  df-ico 13299  df-fz 13457  df-fzo 13604  df-fl 13746  df-ceil 13747  df-limsup 15428  df-liminf 46209

This theorem is referenced by:  liminfreuz  46260