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Theorem liminfreuz 41547
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 2925 . . 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 41546 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙))))
6 breq2 4929 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
76rexbidv 3235 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
87ralbidv 3140 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
9 fveq2 6496 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3349 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
11 liminfreuz.1 . . . . . . . . . . . . 13 𝑗𝐹
12 nfcv 2925 . . . . . . . . . . . . 13 𝑗𝑙
1311, 12nffv 6506 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
14 nfcv 2925 . . . . . . . . . . . 12 𝑗
15 nfcv 2925 . . . . . . . . . . . 12 𝑗𝑥
1613, 14, 15nfbr 4972 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
17 nfv 1874 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
18 fveq2 6496 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq1d 4935 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2016, 17, 19cbvrex 3373 . . . . . . . . . 10 (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2120a1i 11 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2210, 21bitrd 271 . . . . . . . 8 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2322cbvralv 3376 . . . . . . 7 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
2423a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
258, 24bitrd 271 . . . . 5 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
2625cbvrexv 3377 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
27 breq1 4928 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
2827ralbidv 3140 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑙𝑍 𝑥 ≤ (𝐹𝑙)))
2915, 14, 13nfbr 4972 . . . . . . . 8 𝑗 𝑥 ≤ (𝐹𝑙)
30 nfv 1874 . . . . . . . 8 𝑙 𝑥 ≤ (𝐹𝑗)
3118breq2d 4937 . . . . . . . 8 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
3229, 30, 31cbvral 3372 . . . . . . 7 (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3332a1i 11 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑥 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3428, 33bitrd 271 . . . . 5 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗)))
3534cbvrexv 3377 . . . 4 (∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 𝑥 ≤ (𝐹𝑗))
3626, 35anbi12i 618 . . 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 271 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 198  wa 387   = wceq 1508  wcel 2051  wnfc 2909  wral 3081  wrex 3082   class class class wbr 4925  wf 6181  cfv 6185  cr 10332  cle 10473  cz 11791  cuz 12056  lim infclsi 41495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-inf 8700  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-n0 11706  df-z 11792  df-uz 12057  df-q 12161  df-xneg 12322  df-ico 12558  df-fz 12707  df-fzo 12848  df-fl 12975  df-ceil 12976  df-limsup 14687  df-liminf 41496
This theorem is referenced by: (None)
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