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Theorem liminfreuz 45659
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 2904 . . 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 45658 . 2 (𝜑 → ((lim inf‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ∧ ∃𝑦 ∈ ℝ ∀𝑙𝑍 𝑦 ≤ (𝐹𝑙))))
6 breq2 5173 . . . . . . . 8 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
76rexbidv 3181 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
87ralbidv 3180 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
9 fveq2 6919 . . . . . . . . . 10 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
109rexeqdv 3330 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
11 liminfreuz.1 . . . . . . . . . . . . 13 𝑗𝐹
12 nfcv 2904 . . . . . . . . . . . . 13 𝑗𝑙
1311, 12nffv 6929 . . . . . . . . . . . 12 𝑗(𝐹𝑙)
14 nfcv 2904 . . . . . . . . . . . 12 𝑗
15 nfcv 2904 . . . . . . . . . . . 12 𝑗𝑥
1613, 14, 15nfbr 5216 . . . . . . . . . . 11 𝑗(𝐹𝑙) ≤ 𝑥
17 nfv 1913 . . . . . . . . . . 11 𝑙(𝐹𝑗) ≤ 𝑥
18 fveq2 6919 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
1918breq1d 5179 . . . . . . . . . . 11 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
2016, 17, 19cbvrexw 3308 . . . . . . . . . 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 3238 . . . . . . 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 3239 . . . 4 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
27 breq1 5172 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
2827ralbidv 3180 . . . . . 6 (𝑦 = 𝑥 → (∀𝑙𝑍 𝑦 ≤ (𝐹𝑙) ↔ ∀𝑙𝑍 𝑥 ≤ (𝐹𝑙)))
2915, 14, 13nfbr 5216 . . . . . . . 8 𝑗 𝑥 ≤ (𝐹𝑙)
30 nfv 1913 . . . . . . . 8 𝑙 𝑥 ≤ (𝐹𝑗)
3118breq2d 5181 . . . . . . . 8 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
3229, 30, 31cbvralw 3307 . . . . . . 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 3239 . . . 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 2103  wnfc 2888  wral 3063  wrex 3072   class class class wbr 5169  wf 6568  cfv 6572  cr 11179  cle 11321  cz 12635  cuz 12899  lim infclsi 45607
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257  ax-pre-sup 11258
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-isom 6581  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-er 8759  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-sup 9507  df-inf 9508  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-div 11944  df-nn 12290  df-n0 12550  df-z 12636  df-uz 12900  df-q 13010  df-xneg 13171  df-ico 13409  df-fz 13564  df-fzo 13708  df-fl 13839  df-ceil 13840  df-limsup 15513  df-liminf 45608
This theorem is referenced by: (None)
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