Theorem liminfreuzlem 43302

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 1917	. . . . 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 43299	. . . 4 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))))
7	6	eleq1d 2823	. . 3 ⊢ (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗))) ∈ ℝ))
8	4	fvexi 6781	. . . . . . 7 ⊢ 𝑍 ∈ V
9	8	mptex 7092	. . . . . 6 ⊢ (𝑗 ∈ 𝑍 ↦ -(𝐹‘𝑗)) ∈ V
10		limsupcl 15170	. . . . . 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 42969	. . 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	ffvelrnda 6954	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
16	15	renegcld 11390	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → -(𝐹‘𝑗) ∈ ℝ)
17	1, 3, 4, 16	limsupreuzmpt 43239	. . 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 11272	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
19	18	ad2antlr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → -𝑦 ∈ ℝ)
20		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑦 ∈ ℝ)
21	5	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝐹:𝑍⟶ℝ)
22	4	uztrn2 12589	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
23	22	adantll 711	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑗 ∈ 𝑍)
24	21, 23	ffvelrnd 6955	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
25	24	adantllr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
26	20, 25	leneg2d 42947	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝑦 ≤ -(𝐹‘𝑗) ↔ (𝐹‘𝑗) ≤ -𝑦))
27	26	rexbidva 3223	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
28	27	ralbidva 3107	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗)) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦)
31		breq2 5078	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → ((𝐹‘𝑗) ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ -𝑦))
32	31	rexbidv 3224	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
33	32	ralbidv 3108	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ -𝑦))
34	33	rspcev 3560	. . . . . . 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 3214	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥))
37		renegcl 11272	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
38	37	ad2antlr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
39	24	adantllr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → (𝐹‘𝑗) ∈ ℝ)
40		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → 𝑥 ∈ ℝ)
41	39, 40	lenegd 11542	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑘)) → ((𝐹‘𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹‘𝑗)))
42	41	rexbidva 3223	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
43	42	ralbidva 3107	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥) → ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗))
46		breq1 5077	. . . . . . . . . 10 ⊢ (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹‘𝑗) ↔ -𝑥 ≤ -(𝐹‘𝑗)))
47	46	rexbidv 3224	. . . . . . . . 9 ⊢ (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
48	47	ralbidv 3108	. . . . . . . 8 ⊢ (𝑦 = -𝑥 → (∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)𝑦 ≤ -(𝐹‘𝑗) ↔ ∀𝑘 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑘)-𝑥 ≤ -(𝐹‘𝑗)))
49	48	rspcev 3560	. . . . . . 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 3214	. . . . 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 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
54	15	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
55		simplr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑦 ∈ ℝ)
56	54, 55	leneg3d 42956	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (-(𝐹‘𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹‘𝑗)))
57	56	ralbidva 3107	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦) → ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗))
60		breq1 5077	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (𝑥 ≤ (𝐹‘𝑗) ↔ -𝑦 ≤ (𝐹‘𝑗)))
61	60	ralbidv 3108	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → (∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗) ↔ ∀𝑗 ∈ 𝑍 -𝑦 ≤ (𝐹‘𝑗)))
62	61	rspcev 3560	. . . . . . 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 3214	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)))
65	37	ad2antlr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → -𝑥 ∈ ℝ)
66		simplr 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑥 ∈ ℝ)
67	15	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℝ)
68	66, 67	lenegd 11542	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝑥 ≤ (𝐹‘𝑗) ↔ -(𝐹‘𝑗) ≤ -𝑥))
69	68	ralbidva 3107	. . . . . . . . 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 407	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑗 ∈ 𝑍 𝑥 ≤ (𝐹‘𝑗)) → ∀𝑗 ∈ 𝑍 -(𝐹‘𝑗) ≤ -𝑥)
72		brralrspcev 5134	. . . . . . 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 3214	. . . . 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 631	. . 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 396   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  ∃wrex 3065  Vcvv 3430   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6423  ‘cfv 6427  ℝcr 10858  ℝ^*cxr 10996   ≤ cle 10998  -cneg 11194  ℤcz 12307  ℤ_≥cuz 12570  -_𝑒cxne 12833  lim supclsp 15167  lim infclsi 43251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pow 5287  ax-pr 5351  ax-un 7579  ax-cnex 10915  ax-resscn 10916  ax-1cn 10917  ax-icn 10918  ax-addcl 10919  ax-addrcl 10920  ax-mulcl 10921  ax-mulrcl 10922  ax-mulcom 10923  ax-addass 10924  ax-mulass 10925  ax-distr 10926  ax-i2m1 10927  ax-1ne0 10928  ax-1rid 10929  ax-rnegex 10930  ax-rrecex 10931  ax-cnre 10932  ax-pre-lttri 10933  ax-pre-lttrn 10934  ax-pre-ltadd 10935  ax-pre-mulgt0 10936  ax-pre-sup 10937

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5485  df-eprel 5491  df-po 5499  df-so 5500  df-fr 5540  df-we 5542  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-pred 6196  df-ord 6263  df-on 6264  df-lim 6265  df-suc 6266  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-isom 6436  df-riota 7225  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7704  df-1st 7821  df-2nd 7822  df-frecs 8085  df-wrecs 8116  df-recs 8190  df-rdg 8229  df-1o 8285  df-er 8486  df-en 8722  df-dom 8723  df-sdom 8724  df-fin 8725  df-sup 9189  df-inf 9190  df-pnf 10999  df-mnf 11000  df-xr 11001  df-ltxr 11002  df-le 11003  df-sub 11195  df-neg 11196  df-div 11621  df-nn 11962  df-n0 12222  df-z 12308  df-uz 12571  df-q 12677  df-xneg 12836  df-ico 13073  df-fz 13228  df-fzo 13371  df-fl 13500  df-ceil 13501  df-limsup 15168  df-liminf 43252

This theorem is referenced by:  liminfreuz  43303