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Theorem climliminflimsup2 40359
 Description: A sequence of real numbers converges if and only if its superior limit is real and it is less than or equal to its inferior limit (in such a case, they are actually equal, see liminfgelimsupuz 40338). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
climliminflimsup2.1 (𝜑𝑀 ∈ ℤ)
climliminflimsup2.2 𝑍 = (ℤ𝑀)
climliminflimsup2.3 (𝜑𝐹:𝑍⟶ℝ)
Assertion
Ref Expression
climliminflimsup2 (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))

Proof of Theorem climliminflimsup2
StepHypRef Expression
1 climliminflimsup2.1 . . 3 (𝜑𝑀 ∈ ℤ)
2 climliminflimsup2.2 . . 3 𝑍 = (ℤ𝑀)
3 climliminflimsup2.3 . . 3 (𝜑𝐹:𝑍⟶ℝ)
41, 2, 3climliminflimsup 40358 . 2 (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
51adantr 480 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ)
63adantr 480 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ)
7 simprl 809 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
8 simprr 811 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
95, 2, 6, 7, 8liminflimsupclim 40357 . . . . . 6 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ )
101adantr 480 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ)
113adantr 480 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ)
12 simpr 476 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
1310, 2, 11, 12climliminflimsupd 40351 . . . . . . 7 ((𝜑𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹))
1413eqcomd 2657 . . . . . 6 ((𝜑𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹))
159, 14syldan 486 . . . . 5 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) = (lim inf‘𝐹))
1615, 7eqeltrd 2730 . . . 4 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
1716, 8jca 553 . . 3 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
18 simpr 476 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
191adantr 480 . . . . . . . 8 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝑀 ∈ ℤ)
203adantr 480 . . . . . . . 8 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝐹:𝑍⟶ℝ)
2119, 2, 20liminfgelimsupuz 40338 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
2218, 21mpbid 222 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹))
2322adantrl 752 . . . . 5 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) = (lim sup‘𝐹))
24 simprl 809 . . . . 5 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
2523, 24eqeltrd 2730 . . . 4 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
26 simprr 811 . . . 4 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
2725, 26jca 553 . . 3 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
2817, 27impbida 895 . 2 (𝜑 → (((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
294, 28bitrd 268 1 (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   class class class wbr 4685  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  ℝcr 9973   ≤ cle 10113  ℤcz 11415  ℤ≥cuz 11725  lim supclsp 14245   ⇝ cli 14259  lim infclsi 40301 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-ioo 12217  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-ceil 12634  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-liminf 40302 This theorem is referenced by:  climliminflimsup4  40361
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