liminflelimsuplem - Mathbox for Glauco Siliprandi

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Theorem liminflelimsuplem 40803

Description: The superior limit is greater than or equal to the inferior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
liminflelimsuplem.1	⊢ (𝜑 → 𝐹 ∈ 𝑉)
liminflelimsuplem.2	⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)

**Assertion**
Ref	Expression
liminflelimsuplem	⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Distinct variable groups: 𝑗,𝐹,𝑘 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑘) 𝑉(𝑗,𝑘)

Proof of Theorem liminflelimsuplem Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 479	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℝ)
2		simpl 476	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ∈ ℝ)
3	1, 2	ifcld 4352	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
4	3	adantll 707	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
5		liminflelimsuplem.2	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
6	5	ad2antrr 719	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
7		oveq1 6913	. . . . . . . . . . . 12 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (𝑘[,)+∞) = (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
8	7	rexeqdv 3358	. . . . . . . . . . 11 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅ ↔ ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅))
9	8	rspcva 3525	. . . . . . . . . 10 ⊢ ((if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ ∧ ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅)
10	4, 6, 9	syl2anc 581	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
11		inss2 4059	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^
12		infxrcl 12452	. . . . . . . . . . . . . 14 ⊢ (((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
13	11, 12	ax-mp 5	. . . . . . . . . . . . 13 ⊢ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
15		inss2 4059	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^
16		infxrcl 12452	. . . . . . . . . . . . . 14 ⊢ (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
18	17	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
19		inss2 4059	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^
20		supxrcl 12434	. . . . . . . . . . . . . 14 ⊢ (((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
21	19, 20	ax-mp 5	. . . . . . . . . . . . 13 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
22	21	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
23		rexr 10403	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → 𝑖 ∈ ℝ^*)
24	23	ad2antrr 719	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ∈ ℝ^*)
25		pnfxr 10411	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ^*
26	25	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → +∞ ∈ ℝ^*)
27	3	rexrd 10407	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
28	27	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
29		icossxr 12547	. . . . . . . . . . . . . . . . . . . 20 ⊢ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) ⊆ ℝ^*
30		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) → 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
31	29, 30	sseldi 3826	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) → 𝑗 ∈ ℝ^*)
32	31	adantl 475	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ ℝ^*)
33		max1 12305	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
34	33	adantr 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
35		simpr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
36	28, 26, 35	icogelbd 40581	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ≤ 𝑗)
37	24, 28, 32, 34, 36	xrletrd 12282	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ 𝑗)
38	24, 26, 37	icossico2 40587	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑖[,)+∞))
39	38	imass2d 40281	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑖[,)+∞)))
40	39	ssrind 4065	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
41	11	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^)
42		infxrss 12458	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ∧ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
43	40, 41, 42	syl2anc 581	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
44	43	adantr 474	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^*, < ))
45		supxrcl 12434	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
46	15, 45	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
47	46	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
48	15	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^*)
49		simpr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅)
50	48, 49	infxrlesupxr 40459	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^*, < ))
51		rexr 10403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℝ^*)
52	51	ad2antlr 720	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ∈ ℝ^*)
53		max2 12307	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
54	53	adantr 474	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
55	52, 28, 32, 54, 36	xrletrd 12282	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ 𝑗)
56	52, 26, 55	icossico2 40587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑙[,)+∞))
57	56	imass2d 40281	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑙[,)+∞)))
58	57	ssrind 4065	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^))
59	19	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^)
60		supxrss 12451	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ∧ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
61	58, 59, 60	syl2anc 581	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
62	61	adantr 474	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
63	18, 47, 22, 50, 62	xrletrd 12282	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
64	14, 18, 22, 44, 63	xrletrd 12282	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
65	64	ad5ant2345 1495	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
66	65	rexlimdva2 3244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → (∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅ → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < )))
67	10, 66	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
68	67	ralrimiva 3176	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
69		nfv 2015	. . . . . . . . 9 ⊢ Ⅎ𝑙𝜑
70		xrltso 12261	. . . . . . . . . . 11 ⊢ < Or ℝ^*
71	70	supex 8639	. . . . . . . . . 10 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
72	71	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ℝ) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V)
73		breq2 4878	. . . . . . . . 9 ⊢ (𝑦 = sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) → (inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )))
74	69, 72, 73	ralrnmpt3 40275	. . . . . . . 8 ⊢ (𝜑 → (∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )))
75	74	adantr 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )))
76	68, 75	mpbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
77		oveq1 6913	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝑙[,)+∞) = (𝑖[,)+∞))
78	77	imaeq2d 5708	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝐹 “ (𝑙[,)+∞)) = (𝐹 “ (𝑖[,)+∞)))
79	78	ineq1d 4041	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
80	79	supeq1d 8622	. . . . . . . . . 10 ⊢ (𝑙 = 𝑖 → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
81	80	cbvmptv 4974	. . . . . . . . 9 ⊢ (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
82	81	rneqi 5585	. . . . . . . 8 ⊢ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
83	82	raleqi 3355	. . . . . . 7 ⊢ (∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
84	83	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦))
85	76, 84	mpbid 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
86		supxrcl 12434	. . . . . . . . . 10 ⊢ (((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
87	11, 86	ax-mp 5	. . . . . . . . 9 ⊢ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
88	87	rgenw 3134	. . . . . . . 8 ⊢ ∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
89		eqid 2826	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
90	89	rnmptss 6642	. . . . . . . 8 ⊢ (∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^* → ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
91	88, 90	ax-mp 5	. . . . . . 7 ⊢ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*
92	91	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
93	13	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
94		infxrgelb 12454	. . . . . 6 ⊢ ((ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^* ∧ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^) → (inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ↔ ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦))
95	92, 93, 94	syl2anc 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → (inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ↔ ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^*, < ) ≤ 𝑦))
96	85, 95	mpbird 249	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
97	96	ralrimiva 3176	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
98		nfv 2015	. . . 4 ⊢ Ⅎ𝑖𝜑
99		nfcv 2970	. . . 4 ⊢ Ⅎ𝑖ℝ
100		nfmpt1 4971	. . . . . 6 ⊢ Ⅎ𝑖(𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
101	100	nfrn 5602	. . . . 5 ⊢ Ⅎ𝑖ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
102		nfcv 2970	. . . . 5 ⊢ Ⅎ𝑖ℝ^*
103		nfcv 2970	. . . . 5 ⊢ Ⅎ𝑖 <
104	101, 102, 103	nfinf 8658	. . . 4 ⊢ Ⅎ𝑖inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < )
105		infxrcl 12452	. . . . . 6 ⊢ (ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^* → inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
106	91, 105	ax-mp 5	. . . . 5 ⊢ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^
107	106	a1i 11	. . . 4 ⊢ (𝜑 → inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
108	98, 99, 104, 93, 107	supxrleubrnmptf 40476	. . 3 ⊢ (𝜑 → (sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ↔ ∀𝑖 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < )))
109	97, 108	mpbird 249	. 2 ⊢ (𝜑 → sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
110		liminflelimsuplem.1	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
111		eqid 2826	. . . 4 ⊢ (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
112	110, 111	liminfvald 40792	. . 3 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
113	110, 89	limsupvald 40783	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
114	112, 113	breq12d 4887	. 2 ⊢ (𝜑 → ((lim inf‘𝐹) ≤ (lim sup‘𝐹) ↔ sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < )))
115	109, 114	mpbird 249	1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∀wral 3118 ∃wrex 3119 Vcvv 3415 ∩ cin 3798 ⊆ wss 3799 ∅c0 4145 ifcif 4307 class class class wbr 4874 ↦ cmpt 4953 ran crn 5344 “ cima 5346 ‘cfv 6124 (class class class)co 6906 supcsup 8616 infcinf 8617 ℝcr 10252 +∞cpnf 10389 ℝ^*cxr 10391 < clt 10392 ≤ cle 10393 [,)cico 12466 lim supclsp 14579 lim infclsi 40779

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-1st 7429 df-2nd 7430 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-ico 12470 df-limsup 14580 df-liminf 40780

This theorem is referenced by: liminflelimsup 40804

W3C validator