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Theorem liminfreuzlem 45758
Description: Given a function on the reals, its inferior limit is real if and only if two condition holds: 1. there is a real number that is greater than or equal to the function, infinitely often; 2. there is a real number that is smaller than or equal to the function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
liminfreuzlem.1 𝑗𝐹
liminfreuzlem.2 (𝜑𝑀 ∈ ℤ)
liminfreuzlem.3 𝑍 = (ℤ𝑀)
liminfreuzlem.4 (𝜑𝐹:𝑍⟶ℝ)
Assertion
Ref Expression
liminfreuzlem (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑀   𝑗,𝑍,𝑘,𝑥   𝜑,𝑗,𝑘,𝑥
Allowed substitution hints:   𝐹(𝑗)   𝑀(𝑥,𝑘)

Proof of Theorem liminfreuzlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfv 1912 . . . . 5 𝑗𝜑
2 liminfreuzlem.1 . . . . 5 𝑗𝐹
3 liminfreuzlem.2 . . . . 5 (𝜑𝑀 ∈ ℤ)
4 liminfreuzlem.3 . . . . 5 𝑍 = (ℤ𝑀)
5 liminfreuzlem.4 . . . . 5 (𝜑𝐹:𝑍⟶ℝ)
61, 2, 3, 4, 5liminfvaluz4 45755 . . . 4 (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))))
76eleq1d 2824 . . 3 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
84fvexi 6921 . . . . . . 7 𝑍 ∈ V
98mptex 7243 . . . . . 6 (𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V
10 limsupcl 15506 . . . . . 6 ((𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V → (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ*)
119, 10ax-mp 5 . . . . 5 (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ*
1211a1i 11 . . . 4 (𝜑 → (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ*)
1312xnegred 45420 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
147, 13bitr4d 282 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
155ffvelcdmda 7104 . . . . 5 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
1615renegcld 11688 . . . 4 ((𝜑𝑗𝑍) → -(𝐹𝑗) ∈ ℝ)
171, 3, 4, 16limsupreuzmpt 45695 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)))
18 renegcl 11570 . . . . . . . 8 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1918ad2antlr 727 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → -𝑦 ∈ ℝ)
20 simpllr 776 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑦 ∈ ℝ)
215ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐹:𝑍⟶ℝ)
224uztrn2 12895 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2322adantll 714 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2421, 23ffvelcdmd 7105 . . . . . . . . . . . . 13 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2524adantllr 719 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2620, 25leneg2d 45398 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑦 ≤ -(𝐹𝑗) ↔ (𝐹𝑗) ≤ -𝑦))
2726rexbidva 3175 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2827ralbidva 3174 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2928biimpd 229 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3029imp 406 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦)
31 breq2 5152 . . . . . . . . . 10 (𝑥 = -𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ -𝑦))
3231rexbidv 3177 . . . . . . . . 9 (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3332ralbidv 3176 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3433rspcev 3622 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3519, 30, 34syl2anc 584 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635rexlimdva2 3155 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
37 renegcl 11570 . . . . . . . 8 (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
3837ad2antlr 727 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
3924adantllr 719 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
40 simpllr 776 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
4139, 40lenegd 11840 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹𝑗)))
4241rexbidva 3175 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4342ralbidva 3174 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4443biimpd 229 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4544imp 406 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗))
46 breq1 5151 . . . . . . . . . 10 (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹𝑗) ↔ -𝑥 ≤ -(𝐹𝑗)))
4746rexbidv 3177 . . . . . . . . 9 (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4847ralbidv 3176 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4948rspcev 3622 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5038, 45, 49syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5150rexlimdva2 3155 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)))
5236, 51impbid 212 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
5318ad2antlr 727 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
5415adantlr 715 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
55 simplr 769 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → 𝑦 ∈ ℝ)
5654, 55leneg3d 45407 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (-(𝐹𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹𝑗)))
5756ralbidva 3174 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5857biimpd 229 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5958imp 406 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗))
60 breq1 5151 . . . . . . . . 9 (𝑥 = -𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ -𝑦 ≤ (𝐹𝑗)))
6160ralbidv 3176 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
6261rspcev 3622 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6353, 59, 62syl2anc 584 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6463rexlimdva2 3155 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
6537ad2antlr 727 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → -𝑥 ∈ ℝ)
66 simplr 769 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → 𝑥 ∈ ℝ)
6715adantlr 715 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6866, 67lenegd 11840 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝑥 ≤ (𝐹𝑗) ↔ -(𝐹𝑗) ≤ -𝑥))
6968ralbidva 3174 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7069biimpd 229 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7170imp 406 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥)
72 brralrspcev 5208 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7365, 71, 72syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7473rexlimdva2 3155 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦))
7564, 74impbid 212 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
7652, 75anbi12d 632 . . 3 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
7717, 76bitrd 279 . 2 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
7814, 77bitrd 279 1 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wnfc 2888  wral 3059  wrex 3068  Vcvv 3478   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  cr 11152  *cxr 11292  cle 11294  -cneg 11491  cz 12611  cuz 12876  -𝑒cxne 13149  lim supclsp 15503  lim infclsi 45707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-xneg 13152  df-ico 13390  df-fz 13545  df-fzo 13692  df-fl 13829  df-ceil 13830  df-limsup 15504  df-liminf 45708
This theorem is referenced by:  liminfreuz  45759
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