liminflelimsuplem - Mathbox for Glauco Siliprandi

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Theorem liminflelimsuplem 45790

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 484	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℝ)
2		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ∈ ℝ)
3	1, 2	ifcld 4572	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
4	3	adantll 714	. . . . . . . . . 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 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
7		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (𝑘[,)+∞) = (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
8	7	rexeqdv 3327	. . . . . . . . . . 11 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅ ↔ ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅))
9	8	rspcva 3620	. . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
11		inss2 4238	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^
12		infxrcl 13375	. . . . . . . . . . . . . 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 4238	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^
16		infxrcl 13375	. . . . . . . . . . . . . 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 4238	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^
20		supxrcl 13357	. . . . . . . . . . . . . 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 11307	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℝ → 𝑖 ∈ ℝ^*)
24	23	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ∈ ℝ^*)
25		pnfxr 11315	. . . . . . . . . . . . . . . . . 18 ⊢ +∞ ∈ ℝ^*
26	25	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → +∞ ∈ ℝ^*)
27	3	rexrd 11311	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
29		icossxr 13472	. . . . . . . . . . . . . . . . . . . 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	sselid 3981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) → 𝑗 ∈ ℝ^*)
32	31	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ ℝ^*)
33		max1 13227	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
34	33	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
35		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
36	28, 26, 35	icogelbd 45571	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ≤ 𝑗)
37	24, 28, 32, 34, 36	xrletrd 13204	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ 𝑗)
38	24, 26, 37	icossico2 45577	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑖[,)+∞))
39	38	imass2d 45268	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑖[,)+∞)))
40	39	ssrind 4244	. . . . . . . . . . . . . 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 13381	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
44	43	adantr 480	. . . . . . . . . . . 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 13357	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅)
50	48, 49	infxrlesupxr 45447	. . . . . . . . . . . . 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 11307	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℝ^*)
52	51	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ∈ ℝ^*)
53		max2 13229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
55	52, 28, 32, 54, 36	xrletrd 13204	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ 𝑗)
56	52, 26, 55	icossico2 45577	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑙[,)+∞))
57	56	imass2d 45268	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑙[,)+∞)))
58	57	ssrind 4244	. . . . . . . . . . . . . . 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 13374	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
62	61	adantr 480	. . . . . . . . . . . . 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 13204	. . . . . . . . . . . 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 13204	. . . . . . . . . . 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 1372	. . . . . . . . . 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 3157	. . . . . . . . 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 3146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
69		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑙𝜑
70		xrltso 13183	. . . . . . . . . . 11 ⊢ < Or ℝ^*
71	70	supex 9503	. . . . . . . . . 10 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
72	71	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ℝ) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V)
73		breq2 5147	. . . . . . . . 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 45266	. . . . . . . 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 480	. . . . . . 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 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
77		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑖 → (𝑙[,)+∞) = (𝑖[,)+∞))
78	77	imaeq2d 6078	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝐹 “ (𝑙[,)+∞)) = (𝐹 “ (𝑖[,)+∞)))
79	78	ineq1d 4219	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
80	79	supeq1d 9486	. . . . . . . . . 10 ⊢ (𝑙 = 𝑖 → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
81	80	cbvmptv 5255	. . . . . . . . 9 ⊢ (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
82	81	rneqi 5948	. . . . . . . 8 ⊢ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
83	82	raleqi 3324	. . . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
86		supxrcl 13357	. . . . . . . . . 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 3065	. . . . . . . 8 ⊢ ∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
89		eqid 2737	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
90	89	rnmptss 7143	. . . . . . . 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 13377	. . . . . 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 584	. . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
97	96	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
98		nfv 1914	. . . 4 ⊢ Ⅎ𝑖𝜑
99		nfcv 2905	. . . 4 ⊢ Ⅎ𝑖ℝ
100		nfmpt1 5250	. . . . . 6 ⊢ Ⅎ𝑖(𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
101	100	nfrn 5963	. . . . 5 ⊢ Ⅎ𝑖ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
102		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑖ℝ^*
103		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑖 <
104	101, 102, 103	nfinf 9522	. . . 4 ⊢ Ⅎ𝑖inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < )
105		infxrcl 13375	. . . . . 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 45462	. . 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 257	. 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 2737	. . . 4 ⊢ (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
112	110, 111	liminfvald 45779	. . 3 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
113	110, 89	limsupvald 45770	. . 3 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
114	112, 113	breq12d 5156	. 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 257	1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 “ cima 5688 ‘cfv 6561 (class class class)co 7431 supcsup 9480 infcinf 9481 ℝcr 11154 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 lim supclsp 15506 lim infclsi 45766

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-ico 13393 df-limsup 15507 df-liminf 45767

This theorem is referenced by: liminflelimsup 45791

W3C validator