liminflelimsuplem - Mathbox for Glauco Siliprandi

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Theorem liminflelimsuplem 46603

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 4183	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^) ⊆ ℝ^
2		infxrcl 13386	. . . . . . . . . . . . 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 4183	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ℝ^
6		infxrcl 13386	. . . . . . . . . . . . 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 4183	. . . . . . . . . . . . 13 ⊢ ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) ⊆ ℝ^
10	9	supxrcli 46262	. . . . . . . . . . . 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 11279	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℝ → 𝑖 ∈ ℝ^*)
13	12	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ∈ ℝ^*)
14		pnfxr 11287	. . . . . . . . . . . . . . . . 17 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → +∞ ∈ ℝ^*)
16		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ∈ ℝ)
17		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ∈ ℝ)
18	16, 17	ifcld 4529	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
19	18	rexrd 11283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
20	19	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ^*)
21		icossxr 13485	. . . . . . . . . . . . . . . . . . 19 ⊢ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) ⊆ ℝ^*
22	21	sseli 3927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞) → 𝑗 ∈ ℝ^*)
23	22	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ ℝ^*)
24		max1 13237	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
25	24	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
26		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
27	20, 15, 26	icogelbd 13450	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ≤ 𝑗)
28	13, 20, 23, 25, 27	xrletrd 13213	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑖 ≤ 𝑗)
29	13, 15, 28	icossico2d 13474	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑖[,)+∞))
30	29	imass2d 46090	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑖[,)+∞)))
31	30	ssrind 4189	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ⊆ ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
32		infxrss 13392	. . . . . . . . . . . . 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 598	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ))
34	33	adantr 486	. . . . . . . . . . 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 46262	. . . . . . . . . . . . 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 490	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) ∧ ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅) → ((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^) ≠ ∅)
39	37, 38	infxrlesupxr 46264	. . . . . . . . . . . 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 11279	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 ∈ ℝ → 𝑙 ∈ ℝ^*)
41	40	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ∈ ℝ^*)
42		max2 13239	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
43	42	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ if(𝑖 ≤ 𝑙, 𝑙, 𝑖))
44	41, 20, 23, 43, 27	xrletrd 13213	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → 𝑙 ≤ 𝑗)
45	41, 15, 44	icossico2d 13474	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝑗[,)+∞) ⊆ (𝑙[,)+∞))
46	45	imass2d 46090	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → (𝐹 “ (𝑗[,)+∞)) ⊆ (𝐹 “ (𝑙[,)+∞)))
47	46	ssrind 4189	. . . . . . . . . . . . . 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 13385	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ) ∧ 𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)) → sup(((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
50	49	adantr 486	. . . . . . . . . . . 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 13213	. . . . . . . . . . 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 13213	. . . . . . . . . 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 1397	. . . . . . . . 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 7420	. . . . . . . . . . 11 ⊢ (𝑘 = if(𝑖 ≤ 𝑙, 𝑙, 𝑖) → (𝑘[,)+∞) = (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞))
55	54	rexeqdv 3320	. . . . . . . . . 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 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∀𝑘 ∈ ℝ ∃𝑗 ∈ (𝑘[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
58	18	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → if(𝑖 ≤ 𝑙, 𝑙, 𝑖) ∈ ℝ)
59	55, 57, 58	rspcdva 3577	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → ∃𝑗 ∈ (if(𝑖 ≤ 𝑙, 𝑙, 𝑖)[,)+∞)((𝐹 “ (𝑗[,)+∞)) ∩ ℝ^*) ≠ ∅)
60	53, 59	r19.29a 3170	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ ℝ) ∧ 𝑙 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
61	60	ralrimiva 3154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑙 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))
62		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑙𝜑
63		xrltso 13192	. . . . . . . . . . 11 ⊢ < Or ℝ^*
64	63	supex 9434	. . . . . . . . . 10 ⊢ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V
65	64	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑙 ∈ ℝ) → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ V)
66		breq2 5107	. . . . . . . . 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 46088	. . . . . . . 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 486	. . . . . . 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 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
70		oveq1 7420	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑖 → (𝑙[,)+∞) = (𝑖[,)+∞))
71	70	imaeq2d 6056	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑖 → (𝐹 “ (𝑙[,)+∞)) = (𝐹 “ (𝑖[,)+∞)))
72	71	ineq1d 4165	. . . . . . . . . 10 ⊢ (𝑙 = 𝑖 → ((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^) = ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^))
73	72	supeq1d 9416	. . . . . . . . 9 ⊢ (𝑙 = 𝑖 → sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < ) = sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
74	73	cbvmptv 5209	. . . . . . . 8 ⊢ (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
75	74	rneqi 5921	. . . . . . 7 ⊢ ran (𝑙 ∈ ℝ ↦ sup(((𝐹 “ (𝑙[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
76	75	raleqi 3317	. . . . . 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 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → ∀𝑦 ∈ ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ 𝑦)
78	1	supxrcli 46262	. . . . . . . . 9 ⊢ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
79	78	rgenw 3080	. . . . . . . 8 ⊢ ∀𝑖 ∈ ℝ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
80		eqid 2760	. . . . . . . . 9 ⊢ (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
81	80	rnmptss 7116	. . . . . . . 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 13388	. . . . . 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 598	. . . . 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 260	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ ℝ) → inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
87	86	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ ℝ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ) ≤ inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
88		nfv 1947	. . . 4 ⊢ Ⅎ𝑖𝜑
89		nfcv 2922	. . . 4 ⊢ Ⅎ𝑖ℝ
90		nfmpt1 5204	. . . . . 6 ⊢ Ⅎ𝑖(𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
91	90	nfrn 5936	. . . . 5 ⊢ Ⅎ𝑖ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
92		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑖ℝ^*
93		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑖 <
94	91, 92, 93	nfinf 9453	. . . 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 13386	. . . . . 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 46279	. . 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 260	. 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 2760	. . 3 ⊢ (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < ))
103	101, 102	liminfvald 46592	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑖 ∈ ℝ ↦ inf(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
104	101, 80	limsupvald 46583	. 2 ⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
105	100, 103, 104	3brtr4d 5137	1 ⊢ (𝜑 → (lim inf‘𝐹) ≤ (lim sup‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 ifcif 4482 class class class wbr 5103 ↦ cmpt 5186 ran crn 5656 “ cima 5658 ‘cfv 6533 (class class class)co 7413 supcsup 9410 infcinf 9411 ℝcr 11123 +∞cpnf 11264 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 [,)cico 13400 lim supclsp 15557 lim infclsi 46579

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7986 df-2nd 7987 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-ico 13404 df-limsup 15558 df-liminf 46580

This theorem is used by: liminflelimsup 46604

W3C validator