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Theorem liminfreuzlem 42431
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 1915 . . . . 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 42428 . . . 4 (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))))
76eleq1d 2877 . . 3 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
84fvexi 6663 . . . . . . 7 𝑍 ∈ V
98mptex 6967 . . . . . 6 (𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V
10 limsupcl 14825 . . . . . 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 42096 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
147, 13bitr4d 285 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
155ffvelrnda 6832 . . . . 5 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
1615renegcld 11060 . . . 4 ((𝜑𝑗𝑍) → -(𝐹𝑗) ∈ ℝ)
171, 3, 4, 16limsupreuzmpt 42368 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)))
18 renegcl 10942 . . . . . . . 8 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1918ad2antlr 726 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → -𝑦 ∈ ℝ)
20 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑦 ∈ ℝ)
215ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐹:𝑍⟶ℝ)
224uztrn2 12254 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2322adantll 713 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2421, 23ffvelrnd 6833 . . . . . . . . . . . . 13 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2524adantllr 718 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2620, 25leneg2d 42073 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑦 ≤ -(𝐹𝑗) ↔ (𝐹𝑗) ≤ -𝑦))
2726rexbidva 3258 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2827ralbidva 3164 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2928biimpd 232 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3029imp 410 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦)
31 breq2 5037 . . . . . . . . . 10 (𝑥 = -𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ -𝑦))
3231rexbidv 3259 . . . . . . . . 9 (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3332ralbidv 3165 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3433rspcev 3574 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3519, 30, 34syl2anc 587 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635rexlimdva2 3249 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
37 renegcl 10942 . . . . . . . 8 (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
3837ad2antlr 726 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
3924adantllr 718 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
40 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
4139, 40lenegd 11212 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹𝑗)))
4241rexbidva 3258 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4342ralbidva 3164 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4443biimpd 232 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4544imp 410 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗))
46 breq1 5036 . . . . . . . . . 10 (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹𝑗) ↔ -𝑥 ≤ -(𝐹𝑗)))
4746rexbidv 3259 . . . . . . . . 9 (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4847ralbidv 3165 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4948rspcev 3574 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5038, 45, 49syl2anc 587 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5150rexlimdva2 3249 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)))
5236, 51impbid 215 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
5318ad2antlr 726 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
5415adantlr 714 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
55 simplr 768 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → 𝑦 ∈ ℝ)
5654, 55leneg3d 42083 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (-(𝐹𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹𝑗)))
5756ralbidva 3164 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5857biimpd 232 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5958imp 410 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗))
60 breq1 5036 . . . . . . . . 9 (𝑥 = -𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ -𝑦 ≤ (𝐹𝑗)))
6160ralbidv 3165 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
6261rspcev 3574 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6353, 59, 62syl2anc 587 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6463rexlimdva2 3249 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
6537ad2antlr 726 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → -𝑥 ∈ ℝ)
66 simplr 768 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → 𝑥 ∈ ℝ)
6715adantlr 714 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6866, 67lenegd 11212 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝑥 ≤ (𝐹𝑗) ↔ -(𝐹𝑗) ≤ -𝑥))
6968ralbidva 3164 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7069biimpd 232 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7170imp 410 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥)
72 brralrspcev 5093 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7365, 71, 72syl2anc 587 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7473rexlimdva2 3249 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦))
7564, 74impbid 215 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
7652, 75anbi12d 633 . . 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 282 . 2 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
7814, 77bitrd 282 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 209  wa 399   = wceq 1538  wcel 2112  wnfc 2939  wral 3109  wrex 3110  Vcvv 3444   class class class wbr 5033  cmpt 5113  wf 6324  cfv 6328  cr 10529  *cxr 10667  cle 10669  -cneg 10864  cz 11973  cuz 12235  -𝑒cxne 12496  lim supclsp 14822  lim infclsi 42380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-xneg 12499  df-ico 12736  df-fz 12890  df-fzo 13033  df-fl 13161  df-ceil 13162  df-limsup 14823  df-liminf 42381
This theorem is referenced by:  liminfreuz  42432
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