xlimliminflimsup - Mathbox for Glauco Siliprandi

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Theorem xlimliminflimsup 43791

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 723	. . . . 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 723	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
6		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7		xlimdm 43786	. . . . . . 7 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>(~~>‘𝐹))
8	7	biimpi 215	. . . . . 6 ⊢ (𝐹 ∈ dom ~~>* → 𝐹~~>(~~>‘𝐹))
9	8	ad2antlr 724	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹~~>(~~>‘𝐹))
10	2, 3, 5, 6, 9	xlimxrre 43760	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
11	3	eluzelz2 43330	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
12	11	ad2antlr 724	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13		eqid 2736	. . . . . 6 ⊢ (ℤ_≥‘𝑗) = (ℤ_≥‘𝑗)
14		simpr 485	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
15	14	frexr 43311	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ^*)
16	9	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹~~>(~~>‘𝐹))
17	3, 4	fuzxrpmcn 43757	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
18	17	ad3antrrr 727	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
19	11	adantl 482	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
20	18, 19	xlimres 43750	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹))
23		simpllr 773	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
24	12, 13, 15, 22, 23	xlimclimdm 43783	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)) ∈ dom ⇝ )
25	12, 13, 14, 24	climliminflimsupd 43730	. . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
27	17	elexd 3461	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
28	27	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ V)
29	4	fdmd 6663	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
30	26	ssd 43002	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ)
31	29, 30	eqsstrd 3970	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
32	31	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
33	26, 13, 28, 32	liminfresuz2 43716	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim inf‘𝐹))
34	33	eqcomd 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘𝐹) = (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))))
35	34	ad5ant14 755	. . . . 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 43638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘𝐹))
37	36	eqcomd 2742	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘𝐹) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))))
38	37	ad5ant14 755	. . . . 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 2786	. . . 4 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
40	10, 39	rexlimddv2 43752	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41		simpll 764	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝜑)
42	8	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>*‘𝐹))
43		simpr 485	. . . . . . . 8 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → (~~>‘𝐹) = +∞)
44	42, 43	breqtrd 5119	. . . . . . 7 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>+∞)
45	44	adantll 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝐹~~>*+∞)
46	17	liminfcld 43699	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
47	46	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim inf‘𝐹) ∈ ℝ^)
48	17	limsupcld 43619	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
49	48	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim sup‘𝐹) ∈ ℝ^)
50	1, 3, 4	liminflelimsupuz 43714	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
51	50	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
52	49	pnfged 43401	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ +∞)
53	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → 𝑀 ∈ ℤ)
54	4	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹:𝑍⟶ℝ^)
55		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → 𝐹~~>+∞)
56	53, 3, 54, 55	xlimpnfliminf 43789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
57	52, 56	breqtrrd 5121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
58	47, 49, 51, 57	xrletrid 12991	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
59	41, 45, 58	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
60	59	adantlr 712	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
61		simplll 772	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → 𝜑)
62	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>‘𝐹))
63		xlimcl 43751	. . . . . . . . . 10 ⊢ (𝐹~~>(~~>‘𝐹) → (~~>‘𝐹) ∈ ℝ^)
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ~~>* → (~~>‘𝐹) ∈ ℝ^)
65	64	ad2antrr 723	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>‘𝐹) ∈ ℝ^)
66		simplr 766	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67		neqne 2948	. . . . . . . . 9 ⊢ (¬ (~~>‘𝐹) = +∞ → (~~>‘𝐹) ≠ +∞)
68	67	adantl 482	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
69	65, 66, 68	xrnpnfmnf 43402	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
70	62, 69	breqtrd 5119	. . . . . 6 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>*-∞)
71	70	adantlll 715	. . . . 5 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>-∞)
72	46	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim inf‘𝐹) ∈ ℝ^)
73	48	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → (lim sup‘𝐹) ∈ ℝ^)
74	50	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
75	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → 𝑀 ∈ ℤ)
76	4	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹:𝑍⟶ℝ^)
77		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>-∞) → 𝐹~~>-∞)
78	75, 3, 76, 77	xlimmnflimsup 43785	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
79	72	mnfled 43315	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
80	78, 79	eqbrtrd 5115	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
81	72, 73, 74, 80	xrletrid 12991	. . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
82	61, 71, 81	syl2anc 584	. . . 4 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>*‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
83	60, 82	pm2.61dan 810	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ ¬ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
84	40, 83	pm2.61dan 810	. 2 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ~~>*) → (lim inf‘𝐹) = (lim sup‘𝐹))
85	27	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86		mnfxr 11134	. . . . . 6 ⊢ -∞ ∈ ℝ^*
87	86	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → -∞ ∈ ℝ^*)
88		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (lim sup‘𝐹) = -∞)
89	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝑀 ∈ ℤ)
90	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹:𝑍⟶ℝ^*)
91	89, 3, 90	xlimmnflimsup2 43781	. . . . . 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 5855	. . . 4 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
94	93	adantlr 712	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
95	1	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
96	4	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
97		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ)
98	97	renepnfd 11128	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ +∞)
99		simplr 766	. . . . . . . . 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 11129	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
102	95, 3, 96, 98, 101	liminflimsupxrre 43746	. . . . . 6 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
103	3	eluzelz2 43330	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
104	103	ad2antlr 724	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105		eqid 2736	. . . . . . . 8 ⊢ (ℤ_≥‘𝑚) = (ℤ_≥‘𝑚)
106		simpr 485	. . . . . . . 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 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝜑)
108		simpl 483	. . . . . . . . . . . . 13 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
109		simpr 485	. . . . . . . . . . . . 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 750	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
112		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
113	103	3ad2ant3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
114	27	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
115	31	3ad2ant1 1132	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
116	113, 105, 114, 115	liminfresuz2 43716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
117		simp2 1136	. . . . . . . . . . . 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 1370	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
120	119	adantr 481	. . . . . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) = (lim sup‘𝐹))
122	103	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
123	27	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
124	31	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
125	122, 105, 123, 124	liminfresuz2 43716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
126	125	3adant2 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
127	122, 105, 123, 124	limsupresuz2 43638	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
128	127	3adant2 1130	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
129	121, 126, 128	3eqtr4d 2786	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
130	129	ad5ant124 1364	. . . . . . . . 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 43739	. . . . . . . . 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 710	. . . . . . . 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 43780	. . . . . . 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 729	. . . . . . . 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 43788	. . . . . . 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 43752	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138	137	adantlr 712	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
139		simpll 764	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)))
140		simpllr 773	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
141	48	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ∈ ℝ^*)
142		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → ¬ (lim sup‘𝐹) ∈ ℝ)
143		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144	141, 142, 143	xrnmnfpnf 43005	. . . . . . 7 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145	144	adantllr 716	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146	140, 145	eqtrd 2776	. . . . 5 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = +∞)
147	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ V)
148		pnfxr 11131	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ^*)
150	1, 3, 4	xlimpnfliminf2 43790	. . . . . . . 8 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151	150	biimpar 478	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152	147, 149, 151	breldmd 5855	. . . . . 6 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153	152	adantlr 712	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154	139, 146, 153	syl2anc 584	. . . 4 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
155	138, 154	pm2.61dan 810	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) → 𝐹 ∈ dom ~~>*)
156	94, 155	pm2.61dane 3029	. 2 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) → 𝐹 ∈ dom ~~>*)
157	84, 156	impbida 798	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 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ⊆ wss 3898 class class class wbr 5093 dom cdm 5621 ↾ cres 5623 ⟶wf 6476 ‘cfv 6480 (class class class)co 7338 ↑_pm cpm 8688 ℂcc 10971 ℝcr 10972 +∞cpnf 11108 -∞cmnf 11109 ℝ^*cxr 11110 ≤ cle 11112 ℤcz 12421 ℤ_≥cuz 12684 lim supclsp 15279 ⇝ cli 15293 lim infclsi 43680 ~~>*clsxlim 43747

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5230 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 ax-1cn 11031 ax-icn 11032 ax-addcl 11033 ax-addrcl 11034 ax-mulcl 11035 ax-mulrcl 11036 ax-mulcom 11037 ax-addass 11038 ax-mulass 11039 ax-distr 11040 ax-i2m1 11041 ax-1ne0 11042 ax-1rid 11043 ax-rnegex 11044 ax-rrecex 11045 ax-cnre 11046 ax-pre-lttri 11047 ax-pre-lttrn 11048 ax-pre-ltadd 11049 ax-pre-mulgt0 11050 ax-pre-sup 11051

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4854 df-int 4896 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-tr 5211 df-id 5519 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5576 df-we 5578 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6239 df-ord 6306 df-on 6307 df-lim 6308 df-suc 6309 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-f1 6485 df-fo 6486 df-f1o 6487 df-fv 6488 df-isom 6489 df-riota 7294 df-ov 7341 df-oprab 7342 df-mpo 7343 df-om 7782 df-1st 7900 df-2nd 7901 df-frecs 8168 df-wrecs 8199 df-recs 8273 df-rdg 8312 df-1o 8368 df-er 8570 df-map 8689 df-pm 8690 df-en 8806 df-dom 8807 df-sdom 8808 df-fin 8809 df-fi 9269 df-sup 9300 df-inf 9301 df-pnf 11113 df-mnf 11114 df-xr 11115 df-ltxr 11116 df-le 11117 df-sub 11309 df-neg 11310 df-div 11735 df-nn 12076 df-2 12138 df-3 12139 df-4 12140 df-5 12141 df-6 12142 df-7 12143 df-8 12144 df-9 12145 df-n0 12336 df-z 12422 df-dec 12540 df-uz 12685 df-q 12791 df-rp 12833 df-xneg 12950 df-xadd 12951 df-xmul 12952 df-ioo 13185 df-ioc 13186 df-ico 13187 df-icc 13188 df-fz 13342 df-fzo 13485 df-fl 13614 df-ceil 13615 df-seq 13824 df-exp 13885 df-cj 14910 df-re 14911 df-im 14912 df-sqrt 15046 df-abs 15047 df-limsup 15280 df-clim 15297 df-rlim 15298 df-struct 16946 df-slot 16981 df-ndx 16993 df-base 17011 df-plusg 17073 df-mulr 17074 df-starv 17075 df-tset 17079 df-ple 17080 df-ds 17082 df-unif 17083 df-rest 17231 df-topn 17232 df-topgen 17252 df-ordt 17310 df-ps 18382 df-tsr 18383 df-psmet 20696 df-xmet 20697 df-met 20698 df-bl 20699 df-mopn 20700 df-cnfld 20705 df-top 22150 df-topon 22167 df-topsp 22189 df-bases 22203 df-lm 22487 df-haus 22573 df-xms 23580 df-ms 23581 df-liminf 43681 df-xlim 43748

This theorem is referenced by: xlimlimsupleliminf 43792

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