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Theorem liminfreuz 42460
 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
liminfreuz.1 𝑗𝐹
liminfreuz.2 (𝜑𝑀 ∈ ℤ)
liminfreuz.3 𝑍 = (ℤ𝑀)
liminfreuz.4 (𝜑𝐹:𝑍⟶ℝ)
Assertion
Ref Expression
liminfreuz (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑍,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)

Proof of Theorem liminfreuz
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2955 . . 3 𝑙𝐹
2 liminfreuz.2 . . 3 (𝜑𝑀 ∈ ℤ)
3 liminfreuz.3 . . 3 𝑍 = (ℤ𝑀)
4 liminfreuz.4 . . 3 (𝜑𝐹:𝑍⟶ℝ)
51, 2, 3, 4liminfreuzlem 42459 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙))))
6 breq2 5034 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
76rexbidv 3256 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
87ralbidv 3162 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
9 fveq2 6645 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3365 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
11 liminfreuz.1 . . . . . . . . . . . . 13 𝑗𝐹
12 nfcv 2955 . . . . . . . . . . . . 13 𝑗𝑙
1311, 12nffv 6655 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
14 nfcv 2955 . . . . . . . . . . . 12 𝑗
15 nfcv 2955 . . . . . . . . . . . 12 𝑗𝑥
1613, 14, 15nfbr 5077 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
17 nfv 1915 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
18 fveq2 6645 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq1d 5040 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2016, 17, 19cbvrexw 3388 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2210, 21bitrd 282 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2322cbvralvw 3396 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
258, 24bitrd 282 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2625cbvrexvw 3397 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
27 breq1 5033 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
2827ralbidv 3162 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑙𝑍 𝑥 ≤ (𝐹𝑙)))
2915, 14, 13nfbr 5077 . . . . . . . 8 𝑗 𝑥 ≤ (𝐹𝑙)
30 nfv 1915 . . . . . . . 8 𝑙 𝑥 ≤ (𝐹𝑗)
3118breq2d 5042 . . . . . . . 8 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
3229, 30, 31cbvralw 3387 . . . . . . 7 (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3332a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3428, 33bitrd 282 . . . . 5 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3534cbvrexvw 3397 . . . 4 (∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3626, 35anbi12i 629 . . 3 ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3736a1i 11 . 2 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙)) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))))
385, 37bitrd 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 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  ℝcr 10527   ≤ cle 10667  ℤcz 11971  ℤ≥cuz 12233  lim infclsi 42408 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-resscn 10585  ax-1cn 10586  ax-icn 10587  ax-addcl 10588  ax-addrcl 10589  ax-mulcl 10590  ax-mulrcl 10591  ax-mulcom 10592  ax-addass 10593  ax-mulass 10594  ax-distr 10595  ax-i2m1 10596  ax-1ne0 10597  ax-1rid 10598  ax-rnegex 10599  ax-rrecex 10600  ax-cnre 10601  ax-pre-lttri 10602  ax-pre-lttrn 10603  ax-pre-ltadd 10604  ax-pre-mulgt0 10605  ax-pre-sup 10606 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-1st 7673  df-2nd 7674  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-er 8274  df-en 8495  df-dom 8496  df-sdom 8497  df-fin 8498  df-sup 8892  df-inf 8893  df-pnf 10668  df-mnf 10669  df-xr 10670  df-ltxr 10671  df-le 10672  df-sub 10863  df-neg 10864  df-div 11289  df-nn 11628  df-n0 11888  df-z 11972  df-uz 12234  df-q 12339  df-xneg 12497  df-ico 12734  df-fz 12888  df-fzo 13031  df-fl 13159  df-ceil 13160  df-limsup 14822  df-liminf 42409 This theorem is referenced by: (None)
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