Theorem liminfreuzlem 45328

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 1909	. . . . 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 45325	. . . 4 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))))
7	6	eleq1d 2810	. . 3 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
8	4	fvexi 6910	. . . . . . 7 ⊢ 𝑍 ∈ V
9	8	mptex 7235	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V
10		limsupcl 15453	. . . . . 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 44990	. . 3 ⊢ (𝜑 → ((lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
14	7, 13	bitr4d 281	. 2 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
15	5	ffvelcdmda 7093	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
16	15	renegcld 11673	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → -(𝐹‘𝑗) ∈ ℝ)
17	1, 3, 4, 16	limsupreuzmpt 45265	. . 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 11555	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
19	18	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → -𝑦 ∈ ℝ)
20		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑦 ∈ ℝ)
21	5	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶ℝ)
22	4	uztrn2 12874	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
23	22	adantll 712	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
24	21, 23	ffvelcdmd 7094	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
25	24	adantllr 717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
26	20, 25	leneg2d 44968	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑦 ≤ -(𝐹‘𝑗) ↔ (𝐹‘𝑗) ≤ -𝑦))
27	26	rexbidva 3166	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
28	27	ralbidva 3165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
29	28	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
30	29	imp 405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦)
31		breq2 5153	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ -𝑦))
32	31	rexbidv 3168	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
33	32	ralbidv 3167	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
34	33	rspcev 3606	. . . . . . 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 582	. . . . . 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 3146	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37		renegcl 11555	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
38	37	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
39	24	adantllr 717	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
40		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
41	39, 40	lenegd 11825	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹‘𝑗)))
42	41	rexbidva 3166	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
43	42	ralbidva 3165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
44	43	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
45	44	imp 405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗))
46		breq1 5152	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹‘𝑗) ↔ -𝑥 ≤ -(𝐹‘𝑗)))
47	46	rexbidv 3168	. . . . . . . . 9 ⊢ (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
48	47	ralbidv 3167	. . . . . . . 8 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
49	48	rspcev 3606	. . . . . . 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 582	. . . . . 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 3146	. . . . 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 211	. . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
54	15	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
55		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑦 ∈ ℝ)
56	54, 55	leneg3d 44977	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (-(𝐹‘𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹‘𝑗)))
57	56	ralbidva 3165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
58	57	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
59	58	imp 405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗))
60		breq1 5152	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ -𝑦 ≤ (𝐹‘𝑗)))
61	60	ralbidv 3167	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
62	61	rspcev 3606	. . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
63	53, 59, 62	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
64	63	rexlimdva2 3146	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
65	37	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → -𝑥 ∈ ℝ)
66		simplr 767	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
67	15	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67	lenegd 11825	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 ≤ (𝐹‘𝑗) ↔ -(𝐹‘𝑗) ≤ -𝑥))
69	68	ralbidva 3165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
70	69	biimpd 228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥))
71	70	imp 405	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥)
72		brralrspcev 5209	. . . . . . 7 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
73	65, 71, 72	syl2anc 582	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
74	73	rexlimdva2 3146	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦))
75	64, 74	impbid 211	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
76	52, 75	anbi12d 630	. . 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 278	. 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 278	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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2875  ∀wral 3050  ∃wrex 3059  Vcvv 3461   class class class wbr 5149   ↦ cmpt 5232  ⟶wf 6545  ‘cfv 6549  ℝcr 11139  ℝ^*cxr 11279   ≤ cle 11281  -cneg 11477  ℤcz 12591  ℤ_≥cuz 12855  -_𝑒cxne 13124  lim supclsp 15450  lim infclsi 45277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-xneg 13127  df-ico 13365  df-fz 13520  df-fzo 13663  df-fl 13793  df-ceil 13794  df-limsup 15451  df-liminf 45278

This theorem is referenced by:  liminfreuz  45329