xlimliminflimsup - Mathbox for Glauco Siliprandi

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Theorem xlimliminflimsup 43293

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 722	. . . . 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 722	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
6		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7		xlimdm 43288	. . . . . . 7 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>(~~>‘𝐹))
8	7	biimpi 215	. . . . . 6 ⊢ (𝐹 ∈ dom ~~>* → 𝐹~~>(~~>‘𝐹))
9	8	ad2antlr 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹~~>(~~>‘𝐹))
10	2, 3, 5, 6, 9	xlimxrre 43262	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
11	3	eluzelz2 42833	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
12	11	ad2antlr 723	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13		eqid 2738	. . . . . 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 42814	. . . . . . 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 43259	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
18	17	ad3antrrr 726	. . . . . . . . . 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 43252	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹~~>(~~>‘𝐹) ↔ (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹)))
21	16, 20	mpbid 231	. . . . . . . 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 772	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
24	12, 13, 15, 22, 23	xlimclimdm 43285	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)) ∈ dom ⇝ )
25	12, 13, 14, 24	climliminflimsupd 43232	. . . . 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 3442	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ V)
29	4	fdmd 6595	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
30	26	ssd 42519	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ)
31	29, 30	eqsstrd 3955	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
32	31	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
33	26, 13, 28, 32	liminfresuz2 43218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim inf‘𝐹))
34	33	eqcomd 2744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))))
35	34	ad5ant14 754	. . . . 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 43140	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘𝐹))
37	36	eqcomd 2744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
38	37	ad5ant14 754	. . . . 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 2788	. . . 4 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
40	10, 39	rexlimddv2 43254	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41		simpll 763	. . . . . 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 5096	. . . . . . 7 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>+∞)
45	44	adantll 710	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝐹~~>*+∞)
46	17	liminfcld 43201	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
47	46	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim inf‘𝐹) ∈ ℝ^)
48	17	limsupcld 43121	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
49	48	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim sup‘𝐹) ∈ ℝ^)
50	1, 3, 4	liminflelimsupuz 43216	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
51	50	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
52	49	pnfged 42904	. . . . . . . 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 43291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
57	52, 56	breqtrrd 5098	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
58	47, 49, 51, 57	xrletrid 12818	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
59	41, 45, 58	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
60	59	adantlr 711	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61		simplll 771	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
62	8	ad2antrr 722	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>‘𝐹))
63		xlimcl 43253	. . . . . . . . . 10 ⊢ (𝐹~~>(~~>‘𝐹) → (~~>‘𝐹) ∈ ℝ^)
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ~~>* → (~~>‘𝐹) ∈ ℝ^)
65	64	ad2antrr 722	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>‘𝐹) ∈ ℝ^)
66		simplr 765	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67		neqne 2950	. . . . . . . . 9 ⊢ (¬ (~~>‘𝐹) = +∞ → (~~>‘𝐹) ≠ +∞)
68	67	adantl 481	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
69	65, 66, 68	xrnpnfmnf 42905	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
70	62, 69	breqtrd 5096	. . . . . 6 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>*-∞)
71	70	adantlll 714	. . . . 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 43287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
79	72	mnfled 42818	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
80	78, 79	eqbrtrd 5092	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
81	72, 73, 74, 80	xrletrid 12818	. . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
82	61, 71, 81	syl2anc 583	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
83	60, 82	pm2.61dan 809	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
84	40, 83	pm2.61dan 809	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
85	27	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86		mnfxr 10963	. . . . . 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 43283	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
92	88, 91	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
93	85, 87, 92	breldmd 5810	. . . 4 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
94	93	adantlr 711	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
95	1	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
96	4	ad2antrr 722	. . . . . . 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 10957	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99		simplr 765	. . . . . . . . 9 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
100	99, 97	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101	100	renemnfd 10958	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
102	95, 3, 96, 98, 101	liminflimsupxrre 43248	. . . . . 6 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
103	3	eluzelz2 42833	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
104	103	ad2antlr 723	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105		eqid 2738	. . . . . . . 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 771	. . . . . . . . . . 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 2839	. . . . . . . . . . . 12 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111	110	ad4ant23 749	. . . . . . . . . . 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 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
114	27	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
115	31	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
116	113, 105, 114, 115	liminfresuz2 43218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
117		simp2 1135	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
118	116, 117	eqeltrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
119	107, 111, 112, 118	syl3anc 1369	. . . . . . . . . 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 1135	. . . . . . . . . . 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 43218	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
126	125	3adant2 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
127	122, 105, 123, 124	limsupresuz2 43140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
128	127	3adant2 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
129	121, 126, 128	3eqtr4d 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
130	129	ad5ant124 1363	. . . . . . . . 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 43241	. . . . . . . . 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 709	. . . . . . . 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 43282	. . . . . . 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 728	. . . . . . . 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 43290	. . . . . . 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 256	. . . . . 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 43254	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138	137	adantlr 711	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139		simpll 763	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140		simpllr 772	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
141	48	ad2antrr 722	. . . . . . . 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 765	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144	141, 142, 143	xrnmnfpnf 42522	. . . . . . 7 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145	144	adantllr 715	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146	140, 145	eqtrd 2778	. . . . 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 10960	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ^*)
150	1, 3, 4	xlimpnfliminf2 43292	. . . . . . . 8 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151	150	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152	147, 149, 151	breldmd 5810	. . . . . 6 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153	152	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154	139, 146, 153	syl2anc 583	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155	138, 154	pm2.61dan 809	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
156	94, 155	pm2.61dane 3031	. 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
157	84, 156	impbida 797	1 ⊢ (𝜑 → (𝐹 ∈ dom ~~>* ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ⊆ wss 3883 class class class wbr 5070 dom cdm 5580 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_pm cpm 8574 ℂcc 10800 ℝcr 10801 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 ≤ cle 10941 ℤcz 12249 ℤ_≥cuz 12511 lim supclsp 15107 ⇝ cli 15121 lim infclsi 43182 ~~>*clsxlim 43249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fi 9100 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-ceil 13441 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-ordt 17129 df-ps 18199 df-tsr 18200 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-lm 22288 df-haus 22374 df-xms 23381 df-ms 23382 df-liminf 43183 df-xlim 43250

This theorem is referenced by: xlimlimsupleliminf 43294

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