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Theorem liminfreuzlem 40546
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 1994 . . . . 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 40543 . . . 4 (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))))
76eleq1d 2834 . . 3 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
84fvexi 6343 . . . . . . 7 𝑍 ∈ V
98mptex 6629 . . . . . 6 (𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V
10 limsupcl 14411 . . . . . 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 40210 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
147, 13bitr4d 271 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
155ffvelrnda 6502 . . . . 5 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
1615renegcld 10658 . . . 4 ((𝜑𝑗𝑍) → -(𝐹𝑗) ∈ ℝ)
171, 3, 4, 16limsupreuzmpt 40483 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)))
18 renegcl 10545 . . . . . . . 8 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1918ad2antlr 698 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → -𝑦 ∈ ℝ)
20 simpllr 752 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑦 ∈ ℝ)
215ad2antrr 697 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐹:𝑍⟶ℝ)
224uztrn2 11905 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2322adantll 685 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2421, 23ffvelrnd 6503 . . . . . . . . . . . . 13 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2524adantllr 690 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2620, 25leneg2d 40186 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑦 ≤ -(𝐹𝑗) ↔ (𝐹𝑗) ≤ -𝑦))
2726rexbidva 3196 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2827ralbidva 3133 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2928biimpd 219 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3029imp 393 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦)
31 breq2 4788 . . . . . . . . . 10 (𝑥 = -𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ -𝑦))
3231rexbidv 3199 . . . . . . . . 9 (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3332ralbidv 3134 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3433rspcev 3458 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3519, 30, 34syl2anc 565 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635rexlimdva2 39853 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
37 renegcl 10545 . . . . . . . 8 (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
3837ad2antlr 698 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
3924adantllr 690 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
40 simpllr 752 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
4139, 40lenegd 10807 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹𝑗)))
4241rexbidva 3196 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4342ralbidva 3133 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4443biimpd 219 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4544imp 393 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗))
46 breq1 4787 . . . . . . . . . 10 (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹𝑗) ↔ -𝑥 ≤ -(𝐹𝑗)))
4746rexbidv 3199 . . . . . . . . 9 (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4847ralbidv 3134 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4948rspcev 3458 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5038, 45, 49syl2anc 565 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5150rexlimdva2 39853 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)))
5236, 51impbid 202 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
5318ad2antlr 698 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
5415adantlr 686 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
55 simplr 744 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → 𝑦 ∈ ℝ)
5654, 55leneg3d 40197 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (-(𝐹𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹𝑗)))
5756ralbidva 3133 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5857biimpd 219 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5958imp 393 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗))
60 breq1 4787 . . . . . . . . 9 (𝑥 = -𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ -𝑦 ≤ (𝐹𝑗)))
6160ralbidv 3134 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
6261rspcev 3458 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6353, 59, 62syl2anc 565 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6463rexlimdva2 39853 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
6537ad2antlr 698 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → -𝑥 ∈ ℝ)
66 simplr 744 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → 𝑥 ∈ ℝ)
6715adantlr 686 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6866, 67lenegd 10807 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝑥 ≤ (𝐹𝑗) ↔ -(𝐹𝑗) ≤ -𝑥))
6968ralbidva 3133 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7069biimpd 219 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7170imp 393 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥)
72 breq2 4788 . . . . . . . . 9 (𝑦 = -𝑥 → (-(𝐹𝑗) ≤ 𝑦 ↔ -(𝐹𝑗) ≤ -𝑥))
7372ralbidv 3134 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7473rspcev 3458 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7565, 71, 74syl2anc 565 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7675rexlimdva2 39853 . . . . 5 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦))
7764, 76impbid 202 . . . 4 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
7852, 77anbi12d 608 . . 3 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
7917, 78bitrd 268 . 2 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
8014, 79bitrd 268 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 196  wa 382   = wceq 1630  wcel 2144  wnfc 2899  wral 3060  wrex 3061  Vcvv 3349   class class class wbr 4784  cmpt 4861  wf 6027  cfv 6031  cr 10136  *cxr 10274  cle 10276  -cneg 10468  cz 11578  cuz 11887  -𝑒cxne 12147  lim supclsp 14408  lim infclsi 40495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-xneg 12150  df-ico 12385  df-fz 12533  df-fzo 12673  df-fl 12800  df-ceil 12801  df-limsup 14409  df-liminf 40496
This theorem is referenced by:  liminfreuz  40547
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