liminflelimsuplem - Mathbox for Glauco Siliprandi

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Theorem liminflelimsuplem 46218

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		inss2 4166	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^
2		infxrcl 13277	. . . . . . . . . . . . 13 ⊢ (((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
3	1, 2	ax-mp 5	. . . . . . . . . . . 12 ⊢ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
4	3	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
5		inss2 4166	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^
6		infxrcl 13277	. . . . . . . . . . . . 13 ⊢ (((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
7	5, 6	ax-mp 5	. . . . . . . . . . . 12 ⊢ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
8	7	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
9		inss2 4166	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^
10	9	supxrcli 45877	. . . . . . . . . . . 12 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
11	10	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
12		rexr 11182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℝ → 𝑖 ∈ ℝ^*)
13	12	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ∈ ℝ^*)
14		pnfxr 11190	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → +∞ ∈ ℝ^*)
16		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℝ)
17		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ∈ ℝ)
18	16, 17	ifcld 4501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
19	18	rexrd 11186	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
20	19	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
21		icossxr 13376	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) ⊆ ℝ^*
22	21	sseli 3911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) → 𝑗 ∈ ℝ^*)
23	22	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ ℝ^*)
24		max1 13128	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
25	24	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
26		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
27	20, 15, 26	icogelbd 13341	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ≤ 𝑗)
28	13, 20, 23, 25, 27	xrletrd 13104	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ 𝑗)
29	13, 15, 28	icossico2d 13365	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑖[,)+∞))
30	29	imass2d 45705	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑖[,)+∞)))
31	30	ssrind 4172	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
32		infxrss 13283	. . . . . . . . . . . . 13 ⊢ ((((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ∧ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
33	31, 1, 32	sylancl 592	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
34	33	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^*, < ))
35	5	supxrcli 45877	. . . . . . . . . . . . 13 ⊢ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
36	35	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^)
37	5	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^*)
38		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅)
39	37, 38	infxrlesupxr 45879	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^*, < ))
40		rexr 11182	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℝ^*)
41	40	ad2antlr 733	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ∈ ℝ^*)
42		max2 13130	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
43	42	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
44	41, 20, 23, 43, 27	xrletrd 13104	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ 𝑗)
45	41, 15, 44	icossico2d 13365	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑙[,)+∞))
46	45	imass2d 45705	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑙[,)+∞)))
47	46	ssrind 4172	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^))
48	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^)
49	47, 48	xrsupssd 13276	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
51	8, 36, 11, 39, 50	xrletrd 13104	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
52	4, 8, 11, 34, 51	xrletrd 13104	. . . . . . . . . 10 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^*, < ))
53	52	ad5ant2345 1378	. . . . . . . . 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^*, < ))
54		oveq1 7363	. . . . . . . . . . 11 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (𝑘[,)+∞) = (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
55	54	rexeqdv 3298	. . . . . . . . . 10 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅ ↔ ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅))
56		liminflelimsuplem.2	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
57	56	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
58	18	adantll 720	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
59	55, 57, 58	rspcdva 3561	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
60	53, 59	r19.29a 3147	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
61	60	ralrimiva 3131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
62		nfv 1921	. . . . . . . . 9 ⊢ Ⅎ𝑙𝜑
63		xrltso 13083	. . . . . . . . . . 11 ⊢ < Or ℝ^*
64	63	supex 9367	. . . . . . . . . 10 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
65	64	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ℝ) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V)
66		breq2 5076	. . . . . . . . 9 ⊢ (𝑦 = sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) → (inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦 ↔ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )))
67	62, 65, 66	ralrnmpt3 45703	. . . . . . . 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^, < )))
68	67	adantr 481	. . . . . . 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^, < )))
69	61, 68	mpbird 258	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
70		oveq1 7363	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑙[,)+∞) = (𝑖[,)+∞))
71	70	imaeq2d 6012	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → (𝐹 “ (𝑙[,)+∞)) = (𝐹 “ (𝑖[,)+∞)))
72	71	ineq1d 4148	. . . . . . . . . 10 ⊢ (𝑙 = 𝑖 → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
73	72	supeq1d 9349	. . . . . . . . 9 ⊢ (𝑙 = 𝑖 → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
74	73	cbvmptv 5176	. . . . . . . 8 ⊢ (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
75	74	rneqi 5879	. . . . . . 7 ⊢ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
76	75	raleqi 3295	. . . . . 6 ⊢ (∀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)
77	69, 76	sylib 219	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
78	1	supxrcli 45877	. . . . . . . . 9 ⊢ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
79	78	rgenw 3057	. . . . . . . 8 ⊢ ∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
80		eqid 2739	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
81	80	rnmptss 7064	. . . . . . . 8 ⊢ (∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^* → ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
82	79, 81	ax-mp 5	. . . . . . 7 ⊢ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*
83	82	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^*)
84		infxrgelb 13279	. . . . . 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))
85	83, 3, 84	sylancl 592	. . . . 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))
86	77, 85	mpbird 258	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
87	86	ralrimiva 3131	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
88		nfv 1921	. . . 4 ⊢ Ⅎ𝑖𝜑
89		nfcv 2901	. . . 4 ⊢ Ⅎ𝑖ℝ
90		nfmpt1 5171	. . . . . 6 ⊢ Ⅎ𝑖(𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
91	90	nfrn 5894	. . . . 5 ⊢ Ⅎ𝑖ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
92		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑖ℝ^*
93		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑖 <
94	91, 92, 93	nfinf 9386	. . . 4 ⊢ Ⅎ𝑖inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < )
95	3	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
96		infxrcl 13277	. . . . . 6 ⊢ (ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) ⊆ ℝ^* → inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
97	82, 96	ax-mp 5	. . . . 5 ⊢ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^
98	97	a1i 11	. . . 4 ⊢ (𝜑 → inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ∈ ℝ^)
99	88, 89, 94, 95, 98	supxrleubrnmptf 45894	. . 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^*, < )))
100	87, 99	mpbird 258	. 2 ⊢ (𝜑 → sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
101		liminflelimsuplem.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
102		eqid 2739	. . 3 ⊢ (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
103	101, 102	liminfvald 46207	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
104	101, 80	limsupvald 46198	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
105	100, 103, 104	3brtr4d 5104	1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 ifcif 4454 class class class wbr 5072 ↦ cmpt 5153 ran crn 5619 “ cima 5621 ‘cfv 6485 (class class class)co 7356 supcsup 9343 infcinf 9344 ℝcr 11028 +∞cpnf 11167 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 [,)cico 13291 lim supclsp 15423 lim infclsi 46194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-ico 13295 df-limsup 15424 df-liminf 46195

This theorem is referenced by: liminflelimsup 46219

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