xlimliminflimsup - Mathbox for Glauco Siliprandi

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Theorem xlimliminflimsup 43032

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 726	. . . . 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 726	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹:𝑍⟶ℝ^*)
6		simpr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (~~>*‘𝐹) ∈ ℝ)
7		xlimdm 43027	. . . . . . 7 ⊢ (𝐹 ∈ dom ~~>* ↔ 𝐹~~>(~~>‘𝐹))
8	7	biimpi 219	. . . . . 6 ⊢ (𝐹 ∈ dom ~~>* → 𝐹~~>(~~>‘𝐹))
9	8	ad2antlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → 𝐹~~>(~~>‘𝐹))
10	2, 3, 5, 6, 9	xlimxrre 43001	. . . 4 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ)
11	3	eluzelz2 42568	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
12	11	ad2antlr 727	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → 𝑗 ∈ ℤ)
13		eqid 2734	. . . . . 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 42549	. . . . . . 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 42998	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℝ^* ↑_pm ℂ))
18	17	ad3antrrr 730	. . . . . . . . . 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 42991	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) → (𝐹~~>(~~>‘𝐹) ↔ (𝐹 ↾ (ℤ_≥‘𝑗))~~>(~~>‘𝐹)))
21	16, 20	mpbid 235	. . . . . . . 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 776	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (~~>*‘𝐹) ∈ ℝ)
24	12, 13, 15, 22, 23	xlimclimdm 43024	. . . . . 6 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (𝐹 ↾ (ℤ_≥‘𝑗)) ∈ dom ⇝ )
25	12, 13, 14, 24	climliminflimsupd 42971	. . . . 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 3421	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
28	27	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐹 ∈ V)
29	4	fdmd 6545	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝑍)
30	26	ssd 42255	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ⊆ ℤ)
31	29, 30	eqsstrd 3929	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 ⊆ ℤ)
32	31	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
33	26, 13, 28, 32	liminfresuz2 42957	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim inf‘𝐹))
34	33	eqcomd 2740	. . . . . 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 42879	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑗))) = (lim sup‘𝐹))
37	36	eqcomd 2740	. . . . . 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 2784	. . . 4 ⊢ (((((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) ∧ 𝑗 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑗)):(ℤ_≥‘𝑗)⟶ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
40	10, 39	rexlimddv2 42993	. . 3 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) ∈ ℝ) → (lim inf‘𝐹) = (lim sup‘𝐹))
41		simpll 767	. . . . . 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 5069	. . . . . . 7 ⊢ ((𝐹 ∈ dom ~~>* ∧ (~~>‘𝐹) = +∞) → 𝐹~~>+∞)
45	44	adantll 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → 𝐹~~>*+∞)
46	17	liminfcld 42940	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ^*)
47	46	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim inf‘𝐹) ∈ ℝ^)
48	17	limsupcld 42860	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ^*)
49	48	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>+∞) → (lim sup‘𝐹) ∈ ℝ^)
50	1, 3, 4	liminflelimsupuz 42955	. . . . . . . 8 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
51	50	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) ≤ (lim sup‘𝐹))
52	49	pnfged 42641	. . . . . . . 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 43030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = +∞)
57	52, 56	breqtrrd 5071	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
58	47, 49, 51, 57	xrletrid 12728	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*+∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
59	41, 45, 58	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝐹 ∈ dom ~~>) ∧ (~~>‘𝐹) = +∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
60	59	adantlr 715	. . . 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 726	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>(~~>‘𝐹))
63		xlimcl 42992	. . . . . . . . . 10 ⊢ (𝐹~~>(~~>‘𝐹) → (~~>‘𝐹) ∈ ℝ^)
64	8, 63	syl 17	. . . . . . . . 9 ⊢ (𝐹 ∈ dom ~~>* → (~~>‘𝐹) ∈ ℝ^)
65	64	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>‘𝐹) ∈ ℝ^)
66		simplr 769	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → ¬ (~~>*‘𝐹) ∈ ℝ)
67		neqne 2943	. . . . . . . . 9 ⊢ (¬ (~~>‘𝐹) = +∞ → (~~>‘𝐹) ≠ +∞)
68	67	adantl 485	. . . . . . . 8 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) ≠ +∞)
69	65, 66, 68	xrnpnfmnf 42642	. . . . . . 7 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → (~~>*‘𝐹) = -∞)
70	62, 69	breqtrd 5069	. . . . . 6 ⊢ (((𝐹 ∈ dom ~~>* ∧ ¬ (~~>‘𝐹) ∈ ℝ) ∧ ¬ (~~>‘𝐹) = +∞) → 𝐹~~>*-∞)
71	70	adantlll 718	. . . . 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 43026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) = -∞)
79	72	mnfled 42553	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → -∞ ≤ (lim inf‘𝐹))
80	78, 79	eqbrtrd 5065	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
81	72, 73, 74, 80	xrletrid 12728	. . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*-∞) → (lim inf‘𝐹) = (lim sup‘𝐹))
82	61, 71, 81	syl2anc 587	. . . 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 484	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ V)
86		mnfxr 10873	. . . . . 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 43022	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞))
92	88, 91	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹~~>*-∞)
93	85, 87, 92	breldmd 5770	. . . 4 ⊢ ((𝜑 ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
94	93	adantlr 715	. . 3 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) = -∞) → 𝐹 ∈ dom ~~>*)
95	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝑀 ∈ ℤ)
96	4	ad2antrr 726	. . . . . . 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 10867	. . . . . . 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 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
101	100	renemnfd 10868	. . . . . . 7 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ≠ -∞)
102	95, 3, 96, 98, 101	liminflimsupxrre 42987	. . . . . 6 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → ∃𝑚 ∈ 𝑍 (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ)
103	3	eluzelz2 42568	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ)
104	103	ad2antlr 727	. . . . . . . 8 ⊢ (((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) ∧ 𝑚 ∈ 𝑍) ∧ (𝐹 ↾ (ℤ_≥‘𝑚)):(ℤ_≥‘𝑚)⟶ℝ) → 𝑚 ∈ ℤ)
105		eqid 2734	. . . . . . . 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 775	. . . . . . . . . . 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 2834	. . . . . . . . . . . 12 ⊢ (((lim inf‘𝐹) = (lim sup‘𝐹) ∧ (lim sup‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ)
111	110	ad4ant23 753	. . . . . . . . . . 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 1137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ ℤ)
114	27	3ad2ant1 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → 𝐹 ∈ V)
115	31	3ad2ant1 1135	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → dom 𝐹 ⊆ ℤ)
116	113, 105, 114, 115	liminfresuz2 42957	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
117		simp2 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘𝐹) ∈ ℝ)
118	116, 117	eqeltrd 2834	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) ∈ ℝ ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) ∈ ℝ)
119	107, 111, 112, 118	syl3anc 1373	. . . . . . . . . 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 1139	. . . . . . . . . . 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 42957	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
126	125	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim inf‘𝐹))
127	122, 105, 123, 124	limsupresuz2 42879	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
128	127	3adant2 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘𝐹))
129	121, 126, 128	3eqtr4d 2784	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹) ∧ 𝑚 ∈ 𝑍) → (lim inf‘(𝐹 ↾ (ℤ_≥‘𝑚))) = (lim sup‘(𝐹 ↾ (ℤ_≥‘𝑚))))
130	129	ad5ant124 1367	. . . . . . . . 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 42980	. . . . . . . . 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 713	. . . . . . . 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 43021	. . . . . . 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 732	. . . . . . . 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 43029	. . . . . . 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 260	. . . . . 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 42993	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ∈ ℝ) → 𝐹 ∈ dom ~~>*)
138	137	adantlr 715	. . . 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 726	. . . . . . . 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 769	. . . . . . . 8 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) ≠ -∞)
144	141, 142, 143	xrnmnfpnf 42258	. . . . . . 7 ⊢ (((𝜑 ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
145	144	adantllr 719	. . . . . 6 ⊢ ((((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim sup‘𝐹) ≠ -∞) ∧ ¬ (lim sup‘𝐹) ∈ ℝ) → (lim sup‘𝐹) = +∞)
146	140, 145	eqtrd 2774	. . . . 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 10870	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
149	148	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → +∞ ∈ ℝ^*)
150	1, 3, 4	xlimpnfliminf2 43031	. . . . . . . 8 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ (lim inf‘𝐹) = +∞))
151	150	biimpar 481	. . . . . . 7 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹~~>*+∞)
152	147, 149, 151	breldmd 5770	. . . . . 6 ⊢ ((𝜑 ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
153	152	adantlr 715	. . . . 5 ⊢ (((𝜑 ∧ (lim inf‘𝐹) = (lim sup‘𝐹)) ∧ (lim inf‘𝐹) = +∞) → 𝐹 ∈ dom ~~>*)
154	139, 146, 153	syl2anc 587	. . . 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 3022	. 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 Vcvv 3401 ⊆ wss 3857 class class class wbr 5043 dom cdm 5540 ↾ cres 5542 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ↑_pm cpm 8498 ℂcc 10710 ℝcr 10711 +∞cpnf 10847 -∞cmnf 10848 ℝ^*cxr 10849 ≤ cle 10851 ℤcz 12159 ℤ_≥cuz 12421 lim supclsp 15014 ⇝ cli 15028 lim infclsi 42921 ~~>*clsxlim 42988

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fi 9016 df-sup 9047 df-inf 9048 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-ceil 13351 df-seq 13558 df-exp 13619 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-plusg 16780 df-mulr 16781 df-starv 16782 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-rest 16899 df-topn 16900 df-topgen 16920 df-ordt 16978 df-ps 18044 df-tsr 18045 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-cnfld 20336 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-lm 22098 df-haus 22184 df-xms 23190 df-ms 23191 df-liminf 42922 df-xlim 42989

This theorem is referenced by: xlimlimsupleliminf 43033

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