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Theorem liminfreuzlem 45817
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 1914 . . . . 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 45814 . . . 4 (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))))
76eleq1d 2826 . . 3 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
84fvexi 6920 . . . . . . 7 𝑍 ∈ V
98mptex 7243 . . . . . 6 (𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V
10 limsupcl 15509 . . . . . 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 45481 . . 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 11690 . . . 4 ((𝜑𝑗𝑍) → -(𝐹𝑗) ∈ ℝ)
171, 3, 4, 16limsupreuzmpt 45754 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)))
18 renegcl 11572 . . . . . . . 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 12897 . . . . . . . . . . . . . . 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 45459 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑦 ≤ -(𝐹𝑗) ↔ (𝐹𝑗) ≤ -𝑦))
2726rexbidva 3177 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2827ralbidva 3176 . . . . . . . . 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 5147 . . . . . . . . . 10 (𝑥 = -𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ -𝑦))
3231rexbidv 3179 . . . . . . . . 9 (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3332ralbidv 3178 . . . . . . . 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 3157 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
37 renegcl 11572 . . . . . . . 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 11842 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹𝑗)))
4241rexbidva 3177 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4342ralbidva 3176 . . . . . . . . 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 5146 . . . . . . . . . 10 (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹𝑗) ↔ -𝑥 ≤ -(𝐹𝑗)))
4746rexbidv 3179 . . . . . . . . 9 (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4847ralbidv 3178 . . . . . . . 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 3157 . . . . 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 45468 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (-(𝐹𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹𝑗)))
5756ralbidva 3176 . . . . . . . . 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 5146 . . . . . . . . 9 (𝑥 = -𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ -𝑦 ≤ (𝐹𝑗)))
6160ralbidv 3178 . . . . . . . 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 3157 . . . . 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 11842 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝑥 ≤ (𝐹𝑗) ↔ -(𝐹𝑗) ≤ -𝑥))
6968ralbidva 3176 . . . . . . . . 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 5203 . . . . . . 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 3157 . . . . 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 1540  wcel 2108  wnfc 2890  wral 3061  wrex 3070  Vcvv 3480   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  cr 11154  *cxr 11294  cle 11296  -cneg 11493  cz 12613  cuz 12878  -𝑒cxne 13151  lim supclsp 15506  lim infclsi 45766
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-xneg 13154  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-ceil 13833  df-limsup 15507  df-liminf 45767
This theorem is referenced by:  liminfreuz  45818
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