xlimliminflimsup - Mathbox for Glauco Siliprandi

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Theorem xlimliminflimsup 46397

Description: A sequence of extended reals converges if and only if its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 23-Apr-2023.)

**Hypotheses**
Ref	Expression
xlimliminflimsup.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
xlimliminflimsup.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
xlimliminflimsup.f	⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)

**Assertion**
Ref	Expression
xlimliminflimsup	⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Proof of Theorem xlimliminflimsup Dummy variables 𝑗 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xlimliminflimsup.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2	1	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
3		xlimliminflimsup.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4		xlimliminflimsup.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ^*)
5	4	ad2antrr 736	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
6		simpr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7		xlimdm 46392	. . . . . . 7 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>(~~>‘𝐹))
8	7	biimpi 218	. . . . . 6 ⊢ (𝐹 ∈ dom ~~>* → 𝐹~~>(~~>‘𝐹))
9	8	ad2antlr 737	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹~~>(~~>‘𝐹))
10	2, 3, 5, 6, 9	xlimxrre 46366	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
11	3	eluzelz2 45938	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
12	11	ad2antlr 737	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13		eqid 2761	. . . . . 6 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
14		simpr 488	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
15	14	frexr 45921	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ^*)
16	9	adantr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹~~>(~~>‘𝐹))
17	3, 4	fuzxrpmcn 46363	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
18	17	ad3antrrr 740	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
19	11	adantl 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
20	18, 19	xlimres 46356	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹~~>(~~>‘𝐹) ↔ (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹)))
21	16, 20	mpbid 234	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹))
22	21	adantr 484	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹))
23		simpllr 785	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
24	12, 13, 15, 22, 23	xlimclimdm 46389	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)) ∈ dom ⇝ )
25	12, 13, 14, 24	climliminflimsupd 46336	. . . . 5 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
26	11	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
27	17	elexd 3476	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
28	27	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ V)
29	4	fdmd 6697	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
30	26	ssd 45621	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ)
31	29, 30	eqsstrd 3968	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
32	31	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
33	26, 13, 28, 32	liminfresuz2 46322	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim inf‘𝐹))
34	33	eqcomd 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))))
35	34	ad5ant14 767	. . . . 5 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))))
36	26, 13, 28, 32	limsupresuz2 46244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘𝐹))
37	36	eqcomd 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
38	37	ad5ant14 767	. . . . 5 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
39	25, 35, 38	3eqtr4d 2806	. . . 4 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
40	10, 39	rexlimddv2 46358	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41		simpll 776	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝜑)
42	8	adantr 484	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>*‘𝐹))
43		simpr 488	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → (~~>‘𝐹) = +∞)
44	42, 43	breqtrd 5123	. . . . . . 7 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>+∞)
45	44	adantll 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝐹~~>*+∞)
46	17	liminfcld 46305	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
47	46	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim inf‘𝐹) ∈ ℝ^)
48	17	limsupcld 46225	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
49	48	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim sup‘𝐹) ∈ ℝ^)
50	1, 3, 4	liminflelimsupuz 46320	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
51	50	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
52	49	pnfged 13127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
53	1	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → 𝑀 ∈ ℤ)
54	4	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹:𝑍⟶ℝ^)
55		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹~~>+∞)
56	53, 3, 54, 55	xlimpnfliminf 46395	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
57	52, 56	breqtrrd 5125	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
58	47, 49, 51, 57	xrletrid 13151	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
59	41, 45, 58	syl2anc 593	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
60	59	adantlr 725	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61		simplll 784	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
62	8	ad2antrr 736	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>‘𝐹))
63		xlimcl 46357	. . . . . . . . . 10 ⊢ (𝐹~~>(~~>‘𝐹) → (~~>‘𝐹) ∈ ℝ^)
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ~~>* → (~~>‘𝐹) ∈ ℝ^)
65	64	ad2antrr 736	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>‘𝐹) ∈ ℝ^)
66		simplr 778	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67		neqne 2964	. . . . . . . . 9 ⊢ (¬ (~~>‘𝐹) = +∞ → (~~>‘𝐹) ≠ +∞)
68	67	adantl 485	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
69	65, 66, 68	xrnpnfmnf 46009	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
70	62, 69	breqtrd 5123	. . . . . 6 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>*-∞)
71	70	adantlll 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>-∞)
72	46	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim inf‘𝐹) ∈ ℝ^)
73	48	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim sup‘𝐹) ∈ ℝ^)
74	50	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
75	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → 𝑀 ∈ ℤ)
76	4	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹:𝑍⟶ℝ^)
77		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹~~>-∞)
78	75, 3, 76, 77	xlimmnflimsup 46391	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
79	72	mnfled 13132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
80	78, 79	eqbrtrd 5119	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
81	72, 73, 74, 80	xrletrid 13151	. . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
82	61, 71, 81	syl2anc 593	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
83	60, 82	pm2.61dan 822	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
84	40, 83	pm2.61dan 822	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
85	27	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86		mnfxr 11233	. . . . . 6 ⊢ -∞ ∈ ℝ^*
87	86	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ^*)
88		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
89	1	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
90	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ^*)
91	89, 3, 90	xlimmnflimsup2 46387	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
92	88, 91	mpbird 259	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
93	85, 87, 92	breldmd 5884	. . . 4 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
94	93	adantlr 725	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
95	1	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
96	4	ad2antrr 736	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
97		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
98	97	renepnfd 11227	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99		simplr 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
100	99, 97	eqeltrd 2861	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101	100	renemnfd 11228	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
102	95, 3, 96, 98, 101	liminflimsupxrre 46352	. . . . . 6 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
103	3	eluzelz2 45938	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
104	103	ad2antlr 737	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105		eqid 2761	. . . . . . . 8 ⊢ (ℤ_≥‘𝑚) = (ℤ_≥‘𝑚)
106		simpr 488	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
107		simplll 784	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝜑)
108		simpl 486	. . . . . . . . . . . . 13 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109		simpr 488	. . . . . . . . . . . . 13 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110	108, 109	eqeltrd 2861	. . . . . . . . . . . 12 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111	110	ad4ant23 763	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
112		simpr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
113	103	3ad2ant3 1147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
114	27	3ad2ant1 1145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
115	31	3ad2ant1 1145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
116	113, 105, 114, 115	liminfresuz2 46322	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
117		simp2 1149	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
118	116, 117	eqeltrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
119	107, 111, 112, 118	syl3anc 1389	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
120	119	adantr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
121		simp2 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122	103	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
123	27	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
124	31	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
125	122, 105, 123, 124	liminfresuz2 46322	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
126	125	3adant2 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
127	122, 105, 123, 124	limsupresuz2 46244	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
128	127	3adant2 1143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
129	121, 126, 128	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
130	129	ad5ant124 1379	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
131	104, 105, 106	climliminflimsup3 46345	. . . . . . . . 9 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → ((𝐹 ↾ (ℤ_≥‘𝑚)) ∈ dom ⇝ ↔ ((lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ ∧ (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))))
132	120, 130, 131	mpbir2and 723	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑚)) ∈ dom ⇝ )
133	104, 105, 106, 132	dmclimxlim 46386	. . . . . . 7 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑚)) ∈ dom ~~>*)
134	17	ad4antr 742	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
135	134, 104	xlimresdm 46394	. . . . . . 7 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → (𝐹 ∈ dom ~~>* ↔ (𝐹 ↾ (ℤ_≥‘𝑚)) ∈ dom ~~>*))
136	133, 135	mpbird 259	. . . . . 6 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝐹 ∈ dom ~~>*)
137	102, 136	rexlimddv2 46358	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138	137	adantlr 725	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139		simpll 776	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140		simpllr 785	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
141	48	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ^*)
142		simpr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143		simplr 778	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144	141, 142, 143	xrnmnfpnf 45624	. . . . . . 7 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145	144	adantllr 729	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146	140, 145	eqtrd 2796	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
147	27	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148		pnfxr 11230	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ^*)
150	1, 3, 4	xlimpnfliminf2 46396	. . . . . . . 8 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151	150	biimpar 481	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152	147, 149, 151	breldmd 5884	. . . . . 6 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153	152	adantlr 725	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154	139, 146, 153	syl2anc 593	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155	138, 154	pm2.61dan 822	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
156	94, 155	pm2.61dane 3043	. 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
157	84, 156	impbida 810	1 ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 Vcvv 3453 ⊆ wss 3902 class class class wbr 5097 dom cdm 5643 ↾ cres 5645 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ↑_pm cpm 8803 ℂcc 11065 ℝcr 11066 +∞cpnf 11207 -∞cmnf 11208 ℝ^*cxr 11209 ≤ cle 11211 ℤcz 12562 ℤ_≥cuz 12833 lim supclsp 15488 ⇝ cli 15502 lim infclsi 46286 ~~>*clsxlim 46353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9351 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ioc 13348 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-fl 13796 df-ceil 13797 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-limsup 15489 df-clim 15506 df-rlim 15507 df-struct 17174 df-slot 17209 df-ndx 17221 df-base 17237 df-plusg 17290 df-mulr 17291 df-starv 17292 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-rest 17442 df-topn 17443 df-topgen 17463 df-ordt 17522 df-ps 18589 df-tsr 18590 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-lm 23277 df-haus 23363 df-xms 24368 df-ms 24369 df-liminf 46287 df-xlim 46354

This theorem is referenced by: xlimlimsupleliminf 46398

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