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Theorem liminfreuz 45726
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 2908 . . 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 45725 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙))))
6 breq2 5170 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
76rexbidv 3185 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
87ralbidv 3184 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
9 fveq2 6922 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3335 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
11 liminfreuz.1 . . . . . . . . . . . . 13 𝑗𝐹
12 nfcv 2908 . . . . . . . . . . . . 13 𝑗𝑙
1311, 12nffv 6932 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
14 nfcv 2908 . . . . . . . . . . . 12 𝑗
15 nfcv 2908 . . . . . . . . . . . 12 𝑗𝑥
1613, 14, 15nfbr 5213 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
17 nfv 1913 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
18 fveq2 6922 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq1d 5176 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2016, 17, 19cbvrexw 3313 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2210, 21bitrd 279 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2322cbvralvw 3243 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
258, 24bitrd 279 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2625cbvrexvw 3244 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
27 breq1 5169 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
2827ralbidv 3184 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑙𝑍 𝑥 ≤ (𝐹𝑙)))
2915, 14, 13nfbr 5213 . . . . . . . 8 𝑗 𝑥 ≤ (𝐹𝑙)
30 nfv 1913 . . . . . . . 8 𝑙 𝑥 ≤ (𝐹𝑗)
3118breq2d 5178 . . . . . . . 8 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
3229, 30, 31cbvralw 3312 . . . . . . 7 (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3332a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3428, 33bitrd 279 . . . . 5 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3534cbvrexvw 3244 . . . 4 (∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3626, 35anbi12i 627 . . 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 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 1537  wcel 2108  wnfc 2893  wral 3067  wrex 3076   class class class wbr 5166  wf 6571  cfv 6575  cr 11185  cle 11327  cz 12641  cuz 12905  lim infclsi 45674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263  ax-pre-sup 11264
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-isom 6584  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-er 8765  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-sup 9513  df-inf 9514  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-div 11950  df-nn 12296  df-n0 12556  df-z 12642  df-uz 12906  df-q 13016  df-xneg 13177  df-ico 13415  df-fz 13570  df-fzo 13714  df-fl 13845  df-ceil 13846  df-limsup 15519  df-liminf 45675
This theorem is referenced by: (None)
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