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Theorem climliminflimsup2 44525
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 44504). (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
climliminflimsup2.1 (πœ‘ β†’ 𝑀 ∈ β„€)
climliminflimsup2.2 𝑍 = (β„€β‰₯β€˜π‘€)
climliminflimsup2.3 (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
Assertion
Ref Expression
climliminflimsup2 (πœ‘ β†’ (𝐹 ∈ dom ⇝ ↔ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))))
Proof of Theorem climliminflimsup2
StepHypRef Expression
1 climliminflimsup2.1 . . 3 (πœ‘ β†’ 𝑀 ∈ β„€)
2 climliminflimsup2.2 . . 3 𝑍 = (β„€β‰₯β€˜π‘€)
3 climliminflimsup2.3 . . 3 (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
41, 2, 3climliminflimsup 44524 . 2 (πœ‘ β†’ (𝐹 ∈ dom ⇝ ↔ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))))
51adantr 482 . . . . . . 7 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ 𝑀 ∈ β„€)
63adantr 482 . . . . . . 7 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ 𝐹:π‘βŸΆβ„)
7 simprl 770 . . . . . . 7 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim infβ€˜πΉ) ∈ ℝ)
8 simprr 772 . . . . . . 7 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
95, 2, 6, 7, 8liminflimsupclim 44523 . . . . . 6 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ 𝐹 ∈ dom ⇝ )
101adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝑀 ∈ β„€)
113adantr 482 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝐹:π‘βŸΆβ„)
12 simpr 486 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝐹 ∈ dom ⇝ )
1310, 2, 11, 12climliminflimsupd 44517 . . . . . . 7 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
1413eqcomd 2739 . . . . . 6 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ (lim supβ€˜πΉ) = (lim infβ€˜πΉ))
159, 14syldan 592 . . . . 5 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) = (lim infβ€˜πΉ))
1615, 7eqeltrd 2834 . . . 4 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) ∈ ℝ)
1716, 8jca 513 . . 3 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)))
18 simpr 486 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
191adantr 482 . . . . . . . 8 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ 𝑀 ∈ β„€)
203frexr 44095 . . . . . . . . 9 (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)
2120adantr 482 . . . . . . . 8 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ 𝐹:π‘βŸΆβ„*)
2219, 2, 21liminfgelimsupuz 44504 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ ((lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ) ↔ (lim infβ€˜πΉ) = (lim supβ€˜πΉ)))
2318, 22mpbid 231 . . . . . 6 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
2423adantrl 715 . . . . 5 ((πœ‘ ∧ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
25 simprl 770 . . . . 5 ((πœ‘ ∧ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) ∈ ℝ)
2624, 25eqeltrd 2834 . . . 4 ((πœ‘ ∧ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim infβ€˜πΉ) ∈ ℝ)
27 simprr 772 . . . 4 ((πœ‘ ∧ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
2826, 27jca 513 . . 3 ((πœ‘ ∧ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)))
2917, 28impbida 800 . 2 (πœ‘ β†’ (((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) ↔ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))))
304, 29bitrd 279 1 (πœ‘ β†’ (𝐹 ∈ dom ⇝ ↔ ((lim supβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  dom cdm 5677  βŸΆwf 6540  β€˜cfv 6544  β„cr 11109  β„*cxr 11247   ≀ cle 11249  β„€cz 12558  β„€β‰₯cuz 12822  lim supclsp 15414   ⇝ cli 15428  lim infclsi 44467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-ioo 13328  df-ico 13330  df-fz 13485  df-fzo 13628  df-fl 13757  df-ceil 13758  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-limsup 15415  df-clim 15432  df-rlim 15433  df-liminf 44468
This theorem is referenced by:  climliminflimsup4  44527
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