xlimliminflimsup - Mathbox for Glauco Siliprandi

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Theorem xlimliminflimsup 46220

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 727	. . . . 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 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
6		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7		xlimdm 46215	. . . . . . 7 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>(~~>‘𝐹))
8	7	biimpi 216	. . . . . 6 ⊢ (𝐹 ∈ dom ~~>* → 𝐹~~>(~~>‘𝐹))
9	8	ad2antlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹~~>(~~>‘𝐹))
10	2, 3, 5, 6, 9	xlimxrre 46189	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
11	3	eluzelz2 45761	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
12	11	ad2antlr 728	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13		eqid 2737	. . . . . 6 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
14		simpr 484	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
15	14	frexr 45743	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ^*)
16	9	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹~~>(~~>‘𝐹))
17	3, 4	fuzxrpmcn 46186	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
18	17	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
19	11	adantl 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
20	18, 19	xlimres 46179	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹~~>(~~>‘𝐹) ↔ (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹)))
21	16, 20	mpbid 232	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹))
22	21	adantr 480	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹))
23		simpllr 776	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
24	12, 13, 15, 22, 23	xlimclimdm 46212	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)) ∈ dom ⇝ )
25	12, 13, 14, 24	climliminflimsupd 46159	. . . . 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 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
27	17	elexd 3466	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ V)
29	4	fdmd 6680	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
30	26	ssd 45440	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ)
31	29, 30	eqsstrd 3970	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
33	26, 13, 28, 32	liminfresuz2 46145	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim inf‘𝐹))
34	33	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))))
35	34	ad5ant14 758	. . . . 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 46067	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘𝐹))
37	36	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
38	37	ad5ant14 758	. . . . 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 2782	. . . 4 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
40	10, 39	rexlimddv2 46181	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41		simpll 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝜑)
42	8	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>*‘𝐹))
43		simpr 484	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → (~~>‘𝐹) = +∞)
44	42, 43	breqtrd 5126	. . . . . . 7 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>+∞)
45	44	adantll 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝐹~~>*+∞)
46	17	liminfcld 46128	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim inf‘𝐹) ∈ ℝ^)
48	17	limsupcld 46048	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim sup‘𝐹) ∈ ℝ^)
50	1, 3, 4	liminflelimsupuz 46143	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
51	50	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
52	49	pnfged 13057	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
53	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → 𝑀 ∈ ℤ)
54	4	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹:𝑍⟶ℝ^)
55		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹~~>+∞)
56	53, 3, 54, 55	xlimpnfliminf 46218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
57	52, 56	breqtrrd 5128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
58	47, 49, 51, 57	xrletrid 13081	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
59	41, 45, 58	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
60	59	adantlr 716	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61		simplll 775	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
62	8	ad2antrr 727	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>‘𝐹))
63		xlimcl 46180	. . . . . . . . . 10 ⊢ (𝐹~~>(~~>‘𝐹) → (~~>‘𝐹) ∈ ℝ^)
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ~~>* → (~~>‘𝐹) ∈ ℝ^)
65	64	ad2antrr 727	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>‘𝐹) ∈ ℝ^)
66		simplr 769	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67		neqne 2941	. . . . . . . . 9 ⊢ (¬ (~~>‘𝐹) = +∞ → (~~>‘𝐹) ≠ +∞)
68	67	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
69	65, 66, 68	xrnpnfmnf 45832	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
70	62, 69	breqtrd 5126	. . . . . 6 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>*-∞)
71	70	adantlll 719	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>-∞)
72	46	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim inf‘𝐹) ∈ ℝ^)
73	48	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim sup‘𝐹) ∈ ℝ^)
74	50	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
75	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → 𝑀 ∈ ℤ)
76	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹:𝑍⟶ℝ^)
77		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹~~>-∞)
78	75, 3, 76, 77	xlimmnflimsup 46214	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
79	72	mnfled 13062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
80	78, 79	eqbrtrd 5122	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
81	72, 73, 74, 80	xrletrid 13081	. . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
82	61, 71, 81	syl2anc 585	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
83	60, 82	pm2.61dan 813	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
84	40, 83	pm2.61dan 813	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
85	27	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86		mnfxr 11201	. . . . . 6 ⊢ -∞ ∈ ℝ^*
87	86	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ^*)
88		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
89	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
90	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ^*)
91	89, 3, 90	xlimmnflimsup2 46210	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
92	88, 91	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
93	85, 87, 92	breldmd 5869	. . . 4 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
94	93	adantlr 716	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
95	1	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
96	4	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
97		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
98	97	renepnfd 11195	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99		simplr 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
100	99, 97	eqeltrd 2837	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101	100	renemnfd 11196	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
102	95, 3, 96, 98, 101	liminflimsupxrre 46175	. . . . . 6 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
103	3	eluzelz2 45761	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
104	103	ad2antlr 728	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105		eqid 2737	. . . . . . . 8 ⊢ (ℤ_≥‘𝑚) = (ℤ_≥‘𝑚)
106		simpr 484	. . . . . . . 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 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝜑)
108		simpl 482	. . . . . . . . . . . . 13 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109		simpr 484	. . . . . . . . . . . . 13 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
110	108, 109	eqeltrd 2837	. . . . . . . . . . . 12 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111	110	ad4ant23 754	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
112		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
113	103	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
114	27	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
115	31	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
116	113, 105, 114, 115	liminfresuz2 46145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
117		simp2 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
118	116, 117	eqeltrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
119	107, 111, 112, 118	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
120	119	adantr 480	. . . . . . . . 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 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122	103	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
123	27	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
124	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
125	122, 105, 123, 124	liminfresuz2 46145	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
126	125	3adant2 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
127	122, 105, 123, 124	limsupresuz2 46067	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
128	127	3adant2 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
129	121, 126, 128	3eqtr4d 2782	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
130	129	ad5ant124 1368	. . . . . . . . 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 46168	. . . . . . . . 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 714	. . . . . . . 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 46209	. . . . . . 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 733	. . . . . . . 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 46217	. . . . . . 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 257	. . . . . 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 46181	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138	137	adantlr 716	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139		simpll 767	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140		simpllr 776	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
141	48	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ^*)
142		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144	141, 142, 143	xrnmnfpnf 45443	. . . . . . 7 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145	144	adantllr 720	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146	140, 145	eqtrd 2772	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
147	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148		pnfxr 11198	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ^*)
150	1, 3, 4	xlimpnfliminf2 46219	. . . . . . . 8 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151	150	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152	147, 149, 151	breldmd 5869	. . . . . 6 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153	152	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154	139, 146, 153	syl2anc 585	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155	138, 154	pm2.61dan 813	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
156	94, 155	pm2.61dane 3020	. 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
157	84, 156	impbida 801	1 ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ⊆ wss 3903 class class class wbr 5100 dom cdm 5632 ↾ cres 5634 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ↑_pm cpm 8776 ℂcc 11036 ℝcr 11037 +∞cpnf 11175 -∞cmnf 11176 ℝ^*cxr 11177 ≤ cle 11179 ℤcz 12500 ℤ_≥cuz 12763 lim supclsp 15405 ⇝ cli 15419 lim infclsi 46109 ~~>*clsxlim 46176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-ceil 13725 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-topgen 17375 df-ordt 17434 df-ps 18501 df-tsr 18502 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-lm 23185 df-haus 23271 df-xms 24276 df-ms 24277 df-liminf 46110 df-xlim 46177

This theorem is referenced by: xlimlimsupleliminf 46221

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