liminflimsupclim - Mathbox for Glauco Siliprandi

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Theorem liminflimsupclim 44823

Description: A sequence of real numbers converges if its inferior limit is real, and it is greater than or equal to the superior limit (in such a case, they are actually equal, see liminflelimsupuz 44801). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
liminflimsupclim.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
liminflimsupclim.2	⊢ 𝑍 = (ℤ_≥‘𝑀)
liminflimsupclim.3	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
liminflimsupclim.4	⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ)
liminflimsupclim.5	⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))

**Assertion**
Ref	Expression
liminflimsupclim	⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

Proof of Theorem liminflimsupclim Dummy variables 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		climrel 15442	. . 3 ⊢ Rel ⇝
2	1	a1i 11	. 2 ⊢ (𝜑 → Rel ⇝ )
3		liminflimsupclim.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4		liminflimsupclim.2	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ_≥‘𝑀)
5	4	fvexi 6906	. . . . . . . . . 10 ⊢ 𝑍 ∈ V
6	5	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
7	3, 6	fexd 7232	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
8	7	limsupcld 44706	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
9		liminflimsupclim.4	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ)
10	9	rexrd 11270	. . . . . . 7 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
11		liminflimsupclim.5	. . . . . . 7 ⊢ (𝜑 → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
12		liminflimsupclim.1	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	3	frexr 44395	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
14	12, 4, 13	liminflelimsupuz 44801	. . . . . . 7 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
15	8, 10, 11, 14	xrletrid 13140	. . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) = (lim inf‘𝐹))
16	15, 9	eqeltrd 2831	. . . . 5 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ)
17	16	recnd 11248	. . . 4 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℂ)
18		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝐹
19	12	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
20	3	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐹:𝑍⟶ℝ)
21	9	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (lim inf‘𝐹) ∈ ℝ)
22		simpr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
23	18, 19, 4, 20, 21, 22	liminflt 44821	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥))
24	21	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim inf‘𝐹) ∈ ℝ)
25	3	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝐹:𝑍⟶ℝ)
26	4	uztrn2 12847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
27	26	adantll 710	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
28	25, 27	ffvelcdmd 7088	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
29	28	adantllr 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℝ)
30	22	ad2antrr 722	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
31		rpre 12988	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
32	30, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ)
33	24, 29, 32	ltsubadd2d 11818	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥 ↔ (lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥)))
34	33	bicomd 222	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥) ↔ ((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥))
35	28	recnd 11248	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (𝐹‘𝑘) ∈ ℂ)
36	15	eqcomd 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹))
37	36, 17	eqeltrd 2831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℂ)
38	37	ad2antrr 722	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim inf‘𝐹) ∈ ℂ)
39	35, 38	negsubdi2d 11593	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → -((𝐹‘𝑘) − (lim inf‘𝐹)) = ((lim inf‘𝐹) − (𝐹‘𝑘)))
40	39	breq1d 5159	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (-((𝐹‘𝑘) − (lim inf‘𝐹)) < 𝑥 ↔ ((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥))
41	40	adantllr 715	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (-((𝐹‘𝑘) − (lim inf‘𝐹)) < 𝑥 ↔ ((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥))
42	41	bicomd 222	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥 ↔ -((𝐹‘𝑘) − (lim inf‘𝐹)) < 𝑥))
43	29, 24	resubcld 11648	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((𝐹‘𝑘) − (lim inf‘𝐹)) ∈ ℝ)
44		ltnegcon1 11721	. . . . . . . . . . . . . 14 ⊢ ((((𝐹‘𝑘) − (lim inf‘𝐹)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-((𝐹‘𝑘) − (lim inf‘𝐹)) < 𝑥 ↔ -𝑥 < ((𝐹‘𝑘) − (lim inf‘𝐹))))
45	43, 32, 44	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (-((𝐹‘𝑘) − (lim inf‘𝐹)) < 𝑥 ↔ -𝑥 < ((𝐹‘𝑘) − (lim inf‘𝐹))))
46	42, 45	bitrd 278	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((lim inf‘𝐹) − (𝐹‘𝑘)) < 𝑥 ↔ -𝑥 < ((𝐹‘𝑘) − (lim inf‘𝐹))))
47	36	oveq2d 7429	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹‘𝑘) − (lim inf‘𝐹)) = ((𝐹‘𝑘) − (lim sup‘𝐹)))
48	47	breq2d 5161	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-𝑥 < ((𝐹‘𝑘) − (lim inf‘𝐹)) ↔ -𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹))))
49	48	ad3antrrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (-𝑥 < ((𝐹‘𝑘) − (lim inf‘𝐹)) ↔ -𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹))))
50	34, 46, 49	3bitrd 304	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥) ↔ -𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹))))
51	50	ralbidva 3173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹))))
52	51	rexbidva 3174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(lim inf‘𝐹) < ((𝐹‘𝑘) + 𝑥) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹))))
53	23, 52	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)))
54	16	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (lim sup‘𝐹) ∈ ℝ)
55	18, 19, 4, 20, 54, 22	limsupgt 44794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑥) < (lim sup‘𝐹))
56	54	ad2antrr 722	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (lim sup‘𝐹) ∈ ℝ)
57		ltsub23 11700	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (lim sup‘𝐹) ∈ ℝ) → (((𝐹‘𝑘) − 𝑥) < (lim sup‘𝐹) ↔ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
58	29, 32, 56, 57	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) − 𝑥) < (lim sup‘𝐹) ↔ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
59	58	ralbidva 3173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑥) < (lim sup‘𝐹) ↔ ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
60	59	rexbidva 3174	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − 𝑥) < (lim sup‘𝐹) ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
61	55, 60	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)
62	53, 61	jca 510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
63	4	rexanuz2 15302	. . . . . . 7 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥) ↔ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
64	62, 63	sylibr 233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
65		simplll 771	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝜑)
66		simpllr 772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑥 ∈ ℝ⁺)
67	26	adantll 710	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ 𝑍)
68		simpr 483	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) ∧ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)) → (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥))
69	3	ffvelcdmda 7087	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
70	16	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (lim sup‘𝐹) ∈ ℝ)
71	69, 70	resubcld 11648	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (lim sup‘𝐹)) ∈ ℝ)
72	71	adantlr 711	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (lim sup‘𝐹)) ∈ ℝ)
73	31	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) → 𝑥 ∈ ℝ)
74		abslt 15267	. . . . . . . . . . . . 13 ⊢ ((((𝐹‘𝑘) − (lim sup‘𝐹)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥 ↔ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)))
75	72, 73, 74	syl2anc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) → ((abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥 ↔ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)))
76	75	adantr 479	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) ∧ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)) → ((abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥 ↔ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)))
77	68, 76	mpbird 256	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) ∧ (-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥)) → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥)
78	77	ex 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) → ((-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥) → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥))
79	65, 66, 67, 78	syl21anc 834	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → ((-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥) → (abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥))
80	79	ralimdva 3165	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑗 ∈ 𝑍) → (∀𝑘 ∈ (ℤ_≥‘𝑗)(-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥) → ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥))
81	80	reximdva 3166	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(-𝑥 < ((𝐹‘𝑘) − (lim sup‘𝐹)) ∧ ((𝐹‘𝑘) − (lim sup‘𝐹)) < 𝑥) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥))
82	64, 81	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥)
83	82	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥)
84	17, 83	jca 510	. . 3 ⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥))
85		ax-resscn 11171	. . . . . 6 ⊢ ℝ ⊆ ℂ
86	85	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ⊆ ℂ)
87	3, 86	fssd 6736	. . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)
88	18, 12, 4, 87	climuz 44760	. . 3 ⊢ (𝜑 → (𝐹 ⇝ (lim sup‘𝐹) ↔ ((lim sup‘𝐹) ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)(abs‘((𝐹‘𝑘) − (lim sup‘𝐹))) < 𝑥)))
89	84, 88	mpbird 256	. 2 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹))
90		releldm 5944	. 2 ⊢ ((Rel ⇝ ∧ 𝐹 ⇝ (lim sup‘𝐹)) → 𝐹 ∈ dom ⇝ )
91	2, 89, 90	syl2anc 582	1 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ⊆ wss 3949 class class class wbr 5149 dom cdm 5677 Rel wrel 5682 ⟶wf 6540 ‘cfv 6544 (class class class)co 7413 ℂcc 11112 ℝcr 11113 + caddc 11117 < clt 11254 ≤ cle 11255 − cmin 11450 -cneg 11451 ℤcz 12564 ℤ_≥cuz 12828 ℝ⁺crp 12980 abscabs 15187 lim supclsp 15420 ⇝ cli 15434 lim infclsi 44767

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11452 df-neg 11453 df-div 11878 df-nn 12219 df-2 12281 df-3 12282 df-n0 12479 df-z 12565 df-uz 12829 df-q 12939 df-rp 12981 df-xneg 13098 df-xadd 13099 df-ioo 13334 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-ceil 13764 df-seq 13973 df-exp 14034 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-liminf 44768

This theorem is referenced by: climliminflimsup 44824 climliminflimsup2 44825

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