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Theorem liminfreuz 40556
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 2902 . . 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 40555 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙))))
6 breq2 4808 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
76rexbidv 3190 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
87ralbidv 3124 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
9 fveq2 6353 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3284 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
11 liminfreuz.1 . . . . . . . . . . . . 13 𝑗𝐹
12 nfcv 2902 . . . . . . . . . . . . 13 𝑗𝑙
1311, 12nffv 6360 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
14 nfcv 2902 . . . . . . . . . . . 12 𝑗
15 nfcv 2902 . . . . . . . . . . . 12 𝑗𝑥
1613, 14, 15nfbr 4851 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
17 nfv 1992 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
18 fveq2 6353 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq1d 4814 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2016, 17, 19cbvrex 3307 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2210, 21bitrd 268 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2322cbvralv 3310 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
258, 24bitrd 268 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2625cbvrexv 3311 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
27 breq1 4807 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
2827ralbidv 3124 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑙𝑍 𝑥 ≤ (𝐹𝑙)))
2915, 14, 13nfbr 4851 . . . . . . . 8 𝑗 𝑥 ≤ (𝐹𝑙)
30 nfv 1992 . . . . . . . 8 𝑙 𝑥 ≤ (𝐹𝑗)
3118breq2d 4816 . . . . . . . 8 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
3229, 30, 31cbvral 3306 . . . . . . 7 (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3332a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3428, 33bitrd 268 . . . . 5 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3534cbvrexv 3311 . . . 4 (∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3626, 35anbi12i 735 . . 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 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 1632  wcel 2139  wnfc 2889  wral 3050  wrex 3051   class class class wbr 4804  wf 6045  cfv 6049  cr 10147  cle 10287  cz 11589  cuz 11899  lim infclsi 40504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-q 12002  df-xneg 12159  df-ico 12394  df-fz 12540  df-fzo 12680  df-fl 12807  df-ceil 12808  df-limsup 14421  df-liminf 40505
This theorem is referenced by: (None)
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