Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  liminfreuzlem Structured version   Visualization version   GIF version

Theorem liminfreuzlem 40352
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 1883 . . . . 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 40349 . . . 4 (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))))
76eleq1d 2715 . . 3 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ -𝑒(lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ))
84fvexi 6240 . . . . . . 7 𝑍 ∈ V
98mptex 6527 . . . . . 6 (𝑗𝑍 ↦ -(𝐹𝑗)) ∈ V
10 limsupcl 14248 . . . . . 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 40013 . . 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 6399 . . . . 5 ((𝜑𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
1615renegcld 10495 . . . 4 ((𝜑𝑗𝑍) → -(𝐹𝑗) ∈ ℝ)
171, 3, 4, 16limsupreuzmpt 40289 . . 3 (𝜑 → ((lim sup‘(𝑗𝑍 ↦ -(𝐹𝑗))) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ∧ ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)))
18 renegcl 10382 . . . . . . . 8 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
1918ad2antlr 763 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → -𝑦 ∈ ℝ)
20 simpllr 815 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑦 ∈ ℝ)
215ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝐹:𝑍⟶ℝ)
224uztrn2 11743 . . . . . . . . . . . . . . 15 ((𝑘𝑍𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2322adantll 750 . . . . . . . . . . . . . 14 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑗𝑍)
2421, 23ffvelrnd 6400 . . . . . . . . . . . . 13 (((𝜑𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2524adantllr 755 . . . . . . . . . . . 12 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
2620, 25leneg2d 39989 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝑦 ≤ -(𝐹𝑗) ↔ (𝐹𝑗) ≤ -𝑦))
2726rexbidva 3078 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
2827ralbidva 3014 . . . . . . . . 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 444 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦)
31 breq2 4689 . . . . . . . . . 10 (𝑥 = -𝑦 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑗) ≤ -𝑦))
3231rexbidv 3081 . . . . . . . . 9 (𝑥 = -𝑦 → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3332ralbidv 3015 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦))
3433rspcev 3340 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ -𝑦) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3519, 30, 34syl2anc 694 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635rexlimdva2 39653 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) → ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
37 renegcl 10382 . . . . . . . 8 (𝑥 ∈ ℝ → -𝑥 ∈ ℝ)
3837ad2antlr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → -𝑥 ∈ ℝ)
3924adantllr 755 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → (𝐹𝑗) ∈ ℝ)
40 simpllr 815 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → 𝑥 ∈ ℝ)
4139, 40lenegd 10644 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) ∧ 𝑗 ∈ (ℤ𝑘)) → ((𝐹𝑗) ≤ 𝑥 ↔ -𝑥 ≤ -(𝐹𝑗)))
4241rexbidva 3078 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑘𝑍) → (∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4342ralbidva 3014 . . . . . . . . 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 444 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗))
46 breq1 4688 . . . . . . . . . 10 (𝑦 = -𝑥 → (𝑦 ≤ -(𝐹𝑗) ↔ -𝑥 ≤ -(𝐹𝑗)))
4746rexbidv 3081 . . . . . . . . 9 (𝑦 = -𝑥 → (∃𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∃𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4847ralbidv 3015 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)))
4948rspcev 3340 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑥 ≤ -(𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5038, 45, 49syl2anc 694 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) → ∃𝑦 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ -(𝐹𝑗))
5150rexlimdva2 39653 . . . . 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 763 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → -𝑦 ∈ ℝ)
5415adantlr 751 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
55 simplr 807 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → 𝑦 ∈ ℝ)
5654, 55leneg3d 40000 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑗𝑍) → (-(𝐹𝑗) ≤ 𝑦 ↔ -𝑦 ≤ (𝐹𝑗)))
5756ralbidva 3014 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5857biimpd 219 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
5958imp 444 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗))
60 breq1 4688 . . . . . . . . 9 (𝑥 = -𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ -𝑦 ≤ (𝐹𝑗)))
6160ralbidv 3015 . . . . . . . 8 (𝑥 = -𝑦 → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)))
6261rspcev 3340 . . . . . . 7 ((-𝑦 ∈ ℝ ∧ ∀𝑗𝑍 -𝑦 ≤ (𝐹𝑗)) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6353, 59, 62syl2anc 694 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
6463rexlimdva2 39653 . . . . 5 (𝜑 → (∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 → ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
6537ad2antlr 763 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → -𝑥 ∈ ℝ)
66 simplr 807 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → 𝑥 ∈ ℝ)
6715adantlr 751 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝐹𝑗) ∈ ℝ)
6866, 67lenegd 10644 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑗𝑍) → (𝑥 ≤ (𝐹𝑗) ↔ -(𝐹𝑗) ≤ -𝑥))
6968ralbidva 3014 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7069biimpd 219 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑗𝑍 𝑥 ≤ (𝐹𝑗) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7170imp 444 . . . . . . 7 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥)
72 breq2 4689 . . . . . . . . 9 (𝑦 = -𝑥 → (-(𝐹𝑗) ≤ 𝑦 ↔ -(𝐹𝑗) ≤ -𝑥))
7372ralbidv 3015 . . . . . . . 8 (𝑦 = -𝑥 → (∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦 ↔ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥))
7473rspcev 3340 . . . . . . 7 ((-𝑥 ∈ ℝ ∧ ∀𝑗𝑍 -(𝐹𝑗) ≤ -𝑥) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7565, 71, 74syl2anc 694 . . . . . 6 (((𝜑𝑥 ∈ ℝ) ∧ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)) → ∃𝑦 ∈ ℝ ∀𝑗𝑍 -(𝐹𝑗) ≤ 𝑦)
7675rexlimdva2 39653 . . . . 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 747 . . 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 383   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  wf 5922  cfv 5926  cr 9973  *cxr 10111  cle 10113  -cneg 10305  cz 11415  cuz 11725  -𝑒cxne 11981  lim supclsp 14245  lim infclsi 40301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-xneg 11984  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-ceil 12634  df-limsup 14246  df-liminf 40302
This theorem is referenced by:  liminfreuz  40353
  Copyright terms: Public domain W3C validator