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Theorem climliminflimsup2 45120
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 45099). (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 45119 . 2 (πœ‘ β†’ (𝐹 ∈ dom ⇝ ↔ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))))
51adantr 480 . . . . . . 7 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ 𝑀 ∈ β„€)
63adantr 480 . . . . . . 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 45118 . . . . . 6 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ 𝐹 ∈ dom ⇝ )
101adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝑀 ∈ β„€)
113adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝐹:π‘βŸΆβ„)
12 simpr 484 . . . . . . . 8 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ 𝐹 ∈ dom ⇝ )
1310, 2, 11, 12climliminflimsupd 45112 . . . . . . 7 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ (lim infβ€˜πΉ) = (lim supβ€˜πΉ))
1413eqcomd 2733 . . . . . 6 ((πœ‘ ∧ 𝐹 ∈ dom ⇝ ) β†’ (lim supβ€˜πΉ) = (lim infβ€˜πΉ))
159, 14syldan 590 . . . . 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 2828 . . . 4 ((πœ‘ ∧ ((lim infβ€˜πΉ) ∈ ℝ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))) β†’ (lim supβ€˜πΉ) ∈ ℝ)
1716, 8jca 511 . . 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 484 . . . . . . 7 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ))
191adantr 480 . . . . . . . 8 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ 𝑀 ∈ β„€)
203frexr 44690 . . . . . . . . 9 (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)
2120adantr 480 . . . . . . . 8 ((πœ‘ ∧ (lim supβ€˜πΉ) ≀ (lim infβ€˜πΉ)) β†’ 𝐹:π‘βŸΆβ„*)
2219, 2, 21liminfgelimsupuz 45099 . . . . . . 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 2828 . . . 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 511 . . 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 395   = wceq 1534   ∈ wcel 2099   class class class wbr 5142  dom cdm 5672  βŸΆwf 6538  β€˜cfv 6542  β„cr 11129  β„*cxr 11269   ≀ cle 11271  β„€cz 12580  β„€β‰₯cuz 12844  lim supclsp 15438   ⇝ cli 15452  lim infclsi 45062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-xneg 13116  df-xadd 13117  df-ioo 13352  df-ico 13354  df-fz 13509  df-fzo 13652  df-fl 13781  df-ceil 13782  df-seq 13991  df-exp 14051  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-limsup 15439  df-clim 15456  df-rlim 15457  df-liminf 45063
This theorem is referenced by:  climliminflimsup4  45122
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