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Theorem climliminflimsup2 43033
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 43012). (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 43032 . 2 (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
51adantr 484 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ)
63adantr 484 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ)
7 simprl 771 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
8 simprr 773 . . . . . . 7 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
95, 2, 6, 7, 8liminflimsupclim 43031 . . . . . 6 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ )
101adantr 484 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ)
113adantr 484 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ)
12 simpr 488 . . . . . . . 8 ((𝜑𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
1310, 2, 11, 12climliminflimsupd 43025 . . . . . . 7 ((𝜑𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹))
1413eqcomd 2743 . . . . . 6 ((𝜑𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹))
159, 14syldan 594 . . . . 5 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) = (lim inf‘𝐹))
1615, 7eqeltrd 2838 . . . 4 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
1716, 8jca 515 . . 3 ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
18 simpr 488 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
191adantr 484 . . . . . . . 8 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝑀 ∈ ℤ)
203frexr 42605 . . . . . . . . 9 (𝜑𝐹:𝑍⟶ℝ*)
2120adantr 484 . . . . . . . 8 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝐹:𝑍⟶ℝ*)
2219, 2, 21liminfgelimsupuz 43012 . . . . . . 7 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
2318, 22mpbid 235 . . . . . 6 ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹))
2423adantrl 716 . . . . 5 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) = (lim sup‘𝐹))
25 simprl 771 . . . . 5 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
2624, 25eqeltrd 2838 . . . 4 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
27 simprr 773 . . . 4 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
2826, 27jca 515 . . 3 ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
2917, 28impbida 801 . 2 (𝜑 → (((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
304, 29bitrd 282 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 209  wa 399   = wceq 1543  wcel 2110   class class class wbr 5058  dom cdm 5556  wf 6381  cfv 6385  cr 10733  *cxr 10871  cle 10873  cz 12181  cuz 12443  lim supclsp 15036  cli 15050  lim infclsi 42975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5184  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528  ax-cnex 10790  ax-resscn 10791  ax-1cn 10792  ax-icn 10793  ax-addcl 10794  ax-addrcl 10795  ax-mulcl 10796  ax-mulrcl 10797  ax-mulcom 10798  ax-addass 10799  ax-mulass 10800  ax-distr 10801  ax-i2m1 10802  ax-1ne0 10803  ax-1rid 10804  ax-rnegex 10805  ax-rrecex 10806  ax-cnre 10807  ax-pre-lttri 10808  ax-pre-lttrn 10809  ax-pre-ltadd 10810  ax-pre-mulgt0 10811  ax-pre-sup 10812
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3415  df-sbc 3700  df-csb 3817  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-pss 3890  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-tp 4551  df-op 4553  df-uni 4825  df-iun 4911  df-br 5059  df-opab 5121  df-mpt 5141  df-tr 5167  df-id 5460  df-eprel 5465  df-po 5473  df-so 5474  df-fr 5514  df-we 5516  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-rn 5567  df-res 5568  df-ima 5569  df-pred 6165  df-ord 6221  df-on 6222  df-lim 6223  df-suc 6224  df-iota 6343  df-fun 6387  df-fn 6388  df-f 6389  df-f1 6390  df-fo 6391  df-f1o 6392  df-fv 6393  df-isom 6394  df-riota 7175  df-ov 7221  df-oprab 7222  df-mpo 7223  df-om 7650  df-1st 7766  df-2nd 7767  df-wrecs 8052  df-recs 8113  df-rdg 8151  df-1o 8207  df-er 8396  df-pm 8516  df-en 8632  df-dom 8633  df-sdom 8634  df-fin 8635  df-sup 9063  df-inf 9064  df-pnf 10874  df-mnf 10875  df-xr 10876  df-ltxr 10877  df-le 10878  df-sub 11069  df-neg 11070  df-div 11495  df-nn 11836  df-2 11898  df-3 11899  df-n0 12096  df-z 12182  df-uz 12444  df-q 12550  df-rp 12592  df-xneg 12709  df-xadd 12710  df-ioo 12944  df-ico 12946  df-fz 13101  df-fzo 13244  df-fl 13372  df-ceil 13373  df-seq 13580  df-exp 13641  df-cj 14667  df-re 14668  df-im 14669  df-sqrt 14803  df-abs 14804  df-limsup 15037  df-clim 15054  df-rlim 15055  df-liminf 42976
This theorem is referenced by:  climliminflimsup4  43035
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