liminfvalxr - Mathbox for Glauco Siliprandi

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Theorem liminfvalxr 44797

Description: Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
liminfvalxr.1	⊢ Ⅎ𝑥𝐹
liminfvalxr.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
liminfvalxr.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)

**Assertion**
Ref	Expression
liminfvalxr	⊢ (𝜑 → (lim inf‘𝐹) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥))))

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝜑(𝑥) 𝐹(𝑥) 𝑉(𝑥)

Proof of Theorem liminfvalxr Dummy variables 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nftru 1804	. . . . . . 7 ⊢ Ⅎ𝑘⊤
2		inss2 4228	. . . . . . . . 9 ⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
3		infxrcl 13316	. . . . . . . . 9 ⊢ (((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^ → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*
5	4	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℝ) → inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) ∈ ℝ^*)
6	1, 5	supminfxrrnmpt 44479	. . . . . 6 ⊢ (⊤ → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
7	6	mptru 1546	. . . . 5 ⊢ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < )
8	7	a1i 11	. . . 4 ⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
9		tru 1543	. . . . . . . . . . 11 ⊢ ⊤
10		inss2 4228	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^
11	10	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^)
12	11	supminfxr2 44477	. . . . . . . . . . 11 ⊢ (⊤ → sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = -_𝑒inf({𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)}, ℝ^, < ))
13	9, 12	ax-mp 5	. . . . . . . . . 10 ⊢ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = -_𝑒inf({𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)}, ℝ^, < )
14	13	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = -_𝑒inf({𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)}, ℝ^, < ))
15		elinel1 4194	. . . . . . . . . . . . . . . 16 ⊢ (-_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*) → -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)))
16		nfmpt1 5255	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑦(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))
17		xnegex 13191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ -_𝑒(𝐹‘𝑦) ∈ V
18		eqid 2730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) = (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))
19	17, 18	fnmpti 6692	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) Fn 𝐴
20	19	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) Fn 𝐴)
21	20	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞))) → (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) Fn 𝐴)
22		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞))) → -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)))
23	16, 21, 22	fvelimad 6958	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞))) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧)
24	23	3adant2 1129	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^* ∧ -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞))) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧)
25	15, 24	syl3an3 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^* ∧ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*)) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧)
26		elinel2 4195	. . . . . . . . . . . . . . . 16 ⊢ (-_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^) → -_𝑒𝑧 ∈ ℝ^)
27		elinel1 4194	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑦 ∈ 𝐴)
28	17	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) → -_𝑒(𝐹‘𝑦) ∈ V)
29	18	fvmpt2 7008	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ 𝐴 ∧ -_𝑒(𝐹‘𝑦) ∈ V) → ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒(𝐹‘𝑦))
30	27, 28, 29	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) → ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒(𝐹‘𝑦))
31	30	eqcomd 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) → -_𝑒(𝐹‘𝑦) = ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦))
32	31	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) ∧ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧) → -_𝑒(𝐹‘𝑦) = ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦))
33		simpr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) ∧ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧) → ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧)
34	32, 33	eqtrd 2770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) ∧ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧) → -_𝑒(𝐹‘𝑦) = -_𝑒𝑧)
35	34	adantll 710	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧) → -_𝑒(𝐹‘𝑦) = -_𝑒𝑧)
36		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (-_𝑒(𝐹‘𝑦) = -_𝑒𝑧 ↔ -_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
37	36	biimpi 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (-_𝑒(𝐹‘𝑦) = -_𝑒𝑧 → -_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
38	37	adantl 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ -_𝑒(𝐹‘𝑦) = -_𝑒𝑧) → -_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
39		simplr 765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑧 ∈ ℝ^)
40		liminfvalxr.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ^*)
41	40	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝐹:𝐴⟶ℝ^*)
42	27	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑦 ∈ 𝐴)
43	41, 42	ffvelcdmd 7086	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑦) ∈ ℝ^*)
44	43	adantlr 711	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ^) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑦) ∈ ℝ^)
45		xneg11 13198	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) ∈ ℝ^*) → (-_𝑒𝑧 = -_𝑒(𝐹‘𝑦) ↔ 𝑧 = (𝐹‘𝑦)))
46	39, 44, 45	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (-_𝑒𝑧 = -_𝑒(𝐹‘𝑦) ↔ 𝑧 = (𝐹‘𝑦)))
47	46	adantr 479	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ -_𝑒(𝐹‘𝑦) = -_𝑒𝑧) → (-_𝑒𝑧 = -_𝑒(𝐹‘𝑦) ↔ 𝑧 = (𝐹‘𝑦)))
48	38, 47	mpbid 231	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ -_𝑒(𝐹‘𝑦) = -_𝑒𝑧) → 𝑧 = (𝐹‘𝑦))
49	40	ffund 6720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → Fun 𝐹)
50	49, 27	anim12i 611	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (Fun 𝐹 ∧ 𝑦 ∈ 𝐴))
51	50	simpld 493	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → Fun 𝐹)
52	40	fdmd 6727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → dom 𝐹 = 𝐴)
53	52	eqcomd 2736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐴 = dom 𝐹)
54	53	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝐴 = dom 𝐹)
55	42, 54	eleqtrd 2833	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑦 ∈ dom 𝐹)
56	51, 55	jca 510	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹))
57		elinel2 4195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑦 ∈ (𝑘[,)+∞))
58	57	adantl 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑦 ∈ (𝑘[,)+∞))
59		funfvima 7233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑦 ∈ (𝑘[,)+∞) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑘[,)+∞))))
60	56, 58, 59	sylc 65	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑘[,)+∞)))
61	60	ad4ant13 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ -_𝑒(𝐹‘𝑦) = -_𝑒𝑧) → (𝐹‘𝑦) ∈ (𝐹 “ (𝑘[,)+∞)))
62	48, 61	eqeltrd 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ -_𝑒(𝐹‘𝑦) = -_𝑒𝑧) → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞)))
63	35, 62	syldan 589	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ^*) ∧ 𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))) ∧ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧) → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞)))
64	63	rexlimdva2 3155	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^*) → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧 → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞))))
65	64	3adant3 1130	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^* ∧ -_𝑒𝑧 ∈ ℝ^*) → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧 → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞))))
66	26, 65	syl3an3 1163	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^* ∧ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*)) → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))‘𝑦) = -_𝑒𝑧 → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞))))
67	25, 66	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ^* ∧ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*)) → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞)))
68	67	rabssdv 4071	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*)} ⊆ (𝐹 “ (𝑘[,)+∞)))
69		ssrab2 4076	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)} ⊆ ℝ^
70	69	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)} ⊆ ℝ^)
71	68, 70	ssind 4231	. . . . . . . . . . . 12 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)} ⊆ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
72	2	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ ℝ^)
73	40	ffnd 6717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝐴)
74	73	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → 𝐹 Fn 𝐴)
75		elinel1 4194	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*) → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞)))
76	75	adantl 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → 𝑧 ∈ (𝐹 “ (𝑘[,)+∞)))
77		fvelima2 44262	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑧 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑦) = 𝑧)
78	74, 76, 77	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑦) = 𝑧)
79		elinel2 4195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) → 𝑧 ∈ ℝ^)
80		eqcom 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦))
81	80	biimpi 215	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹‘𝑦) = 𝑧 → 𝑧 = (𝐹‘𝑦))
82	81	adantl 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → 𝑧 = (𝐹‘𝑦))
83	82	xnegeqd 44445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → -_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
84		simpl 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → 𝑧 ∈ ℝ^*)
85	82, 84	eqeltrrd 2832	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → (𝐹‘𝑦) ∈ ℝ^*)
86	84, 85, 45	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → (-_𝑒𝑧 = -_𝑒(𝐹‘𝑦) ↔ 𝑧 = (𝐹‘𝑦)))
87	83, 86	mpbid 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → 𝑧 = (𝐹‘𝑦))
88	87	xnegeqd 44445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 ∈ ℝ^* ∧ (𝐹‘𝑦) = 𝑧) → -_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
89	88	ex 411	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℝ^* → ((𝐹‘𝑦) = 𝑧 → -_𝑒𝑧 = -_𝑒(𝐹‘𝑦)))
90	89	reximdv 3168	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ^* → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑦) = 𝑧 → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦)))
91	79, 90	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*) → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑦) = 𝑧 → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦)))
92	91	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → (∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑦) = 𝑧 → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦)))
93	78, 92	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
94		xnegex 13191	. . . . . . . . . . . . . . . 16 ⊢ -_𝑒𝑧 ∈ V
95		elmptima 44260	. . . . . . . . . . . . . . . 16 ⊢ (-_𝑒𝑧 ∈ V → (-_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ↔ ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦)))
96	94, 95	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (-_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ↔ ∃𝑦 ∈ (𝐴 ∩ (𝑘[,)+∞))-_𝑒𝑧 = -_𝑒(𝐹‘𝑦))
97	93, 96	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^*)) → -_𝑒𝑧 ∈ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)))
98	72	sselda 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^)) → 𝑧 ∈ ℝ^)
99	98	xnegcld 13283	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^)) → -_𝑒𝑧 ∈ ℝ^)
100	97, 99	elind 4193	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^)) → -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^))
101	72, 100	ssrabdv 4070	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^) ⊆ {𝑧 ∈ ℝ^ ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^*)})
102	71, 101	eqssd 3998	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)} = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^))
103	102	infeq1d 9474	. . . . . . . . . 10 ⊢ (𝜑 → inf({𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)}, ℝ^, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
104	103	xnegeqd 44445	. . . . . . . . 9 ⊢ (𝜑 → -_𝑒inf({𝑧 ∈ ℝ^* ∣ -_𝑒𝑧 ∈ (((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^)}, ℝ^, < ) = -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
105	14, 104	eqtr2d 2771	. . . . . . . 8 ⊢ (𝜑 → -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ) = sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
106	105	mpteq2dv 5249	. . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
107	106	rneqd 5936	. . . . . 6 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )))
108	107	infeq1d 9474	. . . . 5 ⊢ (𝜑 → inf(ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
109	108	xnegeqd 44445	. . . 4 ⊢ (𝜑 → -_𝑒inf(ran (𝑘 ∈ ℝ ↦ -_𝑒inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
110	8, 109	eqtrd 2770	. . 3 ⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^, < ))
111		liminfvalxr.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
112	40, 111	fexd 7230	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
113		eqid 2730	. . . . 5 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
114	113	liminfval 44773	. . . 4 ⊢ (𝐹 ∈ V → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
115	112, 114	syl 17	. . 3 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
116	111	mptexd 7227	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) ∈ V)
117		eqid 2730	. . . . . 6 ⊢ (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )) = (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < ))
118	117	limsupval 15422	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) ∈ V → (lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
119	116, 118	syl 17	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
120	119	xnegeqd 44445	. . 3 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = -_𝑒inf(ran (𝑘 ∈ ℝ ↦ sup((((𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) “ (𝑘[,)+∞)) ∩ ℝ^), ℝ^, < )), ℝ^*, < ))
121	110, 115, 120	3eqtr4d 2780	. 2 ⊢ (𝜑 → (lim inf‘𝐹) = -_𝑒(lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))))
122		liminfvalxr.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
123		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
124	122, 123	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
125	124	nfxneg 44469	. . . . . 6 ⊢ Ⅎ𝑥-_𝑒(𝐹‘𝑦)
126		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑦-_𝑒(𝐹‘𝑥)
127		fveq2 6890	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
128	127	xnegeqd 44445	. . . . . 6 ⊢ (𝑦 = 𝑥 → -_𝑒(𝐹‘𝑦) = -_𝑒(𝐹‘𝑥))
129	125, 126, 128	cbvmpt 5258	. . . . 5 ⊢ (𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦)) = (𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥))
130	129	fveq2i 6893	. . . 4 ⊢ (lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥)))
131	130	xnegeqi 44448	. . 3 ⊢ -_𝑒(lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥)))
132	131	a1i 11	. 2 ⊢ (𝜑 → -_𝑒(lim sup‘(𝑦 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑦))) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥))))
133	121, 132	eqtrd 2770	1 ⊢ (𝜑 → (lim inf‘𝐹) = -_𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -_𝑒(𝐹‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∈ wcel 2104 Ⅎwnfc 2881 ∃wrex 3068 {crab 3430 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 “ cima 5678 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 supcsup 9437 infcinf 9438 ℝcr 11111 +∞cpnf 11249 ℝ^*cxr 11251 < clt 11252 -_𝑒cxne 13093 [,)cico 13330 lim supclsp 15418 lim infclsi 44765

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-xneg 13096 df-limsup 15419 df-liminf 44766

This theorem is referenced by: liminfvalxrmpt 44800

W3C validator