Theorem liminfreuzlem 45988

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 1915	. . . . 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 45985	. . . 4 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))))
7	6	eleq1d 2819	. . 3 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
8	4	fvexi 6846	. . . . . . 7 ⊢ 𝑍 ∈ V
9	8	mptex 7167	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V
10		limsupcl 15394	. . . . . 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 45656	. . 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 7027	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
16	15	renegcld 11562	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → -(𝐹‘𝑗) ∈ ℝ)
17	1, 3, 4, 16	limsupreuzmpt 45925	. . 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 11442	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
19	18	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → -𝑦 ∈ ℝ)
20		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑦 ∈ ℝ)
21	5	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶ℝ)
22	4	uztrn2 12768	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
23	22	adantll 714	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
24	21, 23	ffvelcdmd 7028	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
25	24	adantllr 719	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
26	20, 25	leneg2d 45634	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑦 ≤ -(𝐹‘𝑗) ↔ (𝐹‘𝑗) ≤ -𝑦))
27	26	rexbidva 3156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
28	27	ralbidva 3155	. . . . . . . . 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 5100	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ -𝑦))
32	31	rexbidv 3158	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
33	32	ralbidv 3157	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
34	33	rspcev 3574	. . . . . . 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 584	. . . . . 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 3137	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37		renegcl 11442	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
38	37	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
39	24	adantllr 719	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
40		simpllr 775	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
41	39, 40	lenegd 11714	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹‘𝑗)))
42	41	rexbidva 3156	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
43	42	ralbidva 3155	. . . . . . . . 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 5099	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹‘𝑗) ↔ -𝑥 ≤ -(𝐹‘𝑗)))
47	46	rexbidv 3158	. . . . . . . . 9 ⊢ (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
48	47	ralbidv 3157	. . . . . . . 8 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
49	48	rspcev 3574	. . . . . . 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 584	. . . . . 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 3137	. . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
54	15	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
55		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑦 ∈ ℝ)
56	54, 55	leneg3d 45643	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (-(𝐹‘𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹‘𝑗)))
57	56	ralbidva 3155	. . . . . . . . 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 5099	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ -𝑦 ≤ (𝐹‘𝑗)))
61	60	ralbidv 3157	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
62	61	rspcev 3574	. . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
63	53, 59, 62	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗))
64	63	rexlimdva2 3137	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
65	37	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → -𝑥 ∈ ℝ)
66		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
67	15	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67	lenegd 11714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 ≤ (𝐹‘𝑗) ↔ -(𝐹‘𝑗) ≤ -𝑥))
69	68	ralbidva 3155	. . . . . . . . 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 5156	. . . . . . 7 ⊢ ((-𝑥 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
73	65, 71, 72	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦)
74	73	rexlimdva2 3137	. . . . 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 632	. . 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 1541   ∈ wcel 2113  Ⅎwnfc 2881  ∀wral 3049  ∃wrex 3058  Vcvv 3438   class class class wbr 5096   ↦ cmpt 5177  ⟶wf 6486  ‘cfv 6490  ℝcr 11023  ℝ^*cxr 11163   ≤ cle 11165  -cneg 11363  ℤcz 12486  ℤ_≥cuz 12749  -_𝑒cxne 13021  lim supclsp 15391  lim infclsi 45937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-sup 9343  df-inf 9344  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-n0 12400  df-z 12487  df-uz 12750  df-q 12860  df-xneg 13024  df-ico 13265  df-fz 13422  df-fzo 13569  df-fl 13710  df-ceil 13711  df-limsup 15392  df-liminf 45938

This theorem is referenced by:  liminfreuz  45989