Theorem climliminflimsup2 46252

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 46231). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
climliminflimsup2.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
climliminflimsup2.2	⊢ 𝑍 = (ℤ≥‘𝑀)
climliminflimsup2.3	⊢ (𝜑 → 𝐹:𝑍⟶ℝ)

Assertion

Ref	Expression
climliminflimsup2	⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))

Proof of Theorem climliminflimsup2

Step	Hyp	Ref	Expression
1		climliminflimsup2.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		climliminflimsup2.2	. . 3 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		climliminflimsup2.3	. . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ)
4	1, 2, 3	climliminflimsup 46251	. 2 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
5	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝑀 ∈ ℤ)
6	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹:𝑍⟶ℝ)
7		simprl 776	. . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
8		simprr 778	. . . . . . 7 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
9	5, 2, 6, 7, 8	liminflimsupclim 46250	. . . . . 6 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → 𝐹 ∈ dom ⇝ )
10	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝑀 ∈ ℤ)
11	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹:𝑍⟶ℝ)
12		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → 𝐹 ∈ dom ⇝ )
13	10, 2, 11, 12	climliminflimsupd 46244	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim inf‘𝐹) = (lim sup‘𝐹))
14	13	eqcomd 2745	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹 ∈ dom ⇝ ) → (lim sup‘𝐹) = (lim inf‘𝐹))
15	9, 14	syldan 597	. . . . 5 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) = (lim inf‘𝐹))
16	15, 7	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
17	16, 8	jca 516	. . 3 ⊢ ((𝜑 ∧ ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
18		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
19	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝑀 ∈ ℤ)
20	3	frexr 45829	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*)
21	20	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → 𝐹:𝑍⟶ℝ*)
22	19, 2, 21	liminfgelimsupuz 46231	. . . . . . 7 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → ((lim sup‘𝐹) ≤ (lim inf‘𝐹) ↔ (lim inf‘𝐹) = (lim sup‘𝐹)))
23	18, 22	mpbid 233	. . . . . 6 ⊢ ((𝜑 ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) → (lim inf‘𝐹) = (lim sup‘𝐹))
24	23	adantrl 722	. . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) = (lim sup‘𝐹))
25		simprl 776	. . . . 5 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ∈ ℝ)
26	24, 25	eqeltrd 2839	. . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim inf‘𝐹) ∈ ℝ)
27		simprr 778	. . . 4 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → (lim sup‘𝐹) ≤ (lim inf‘𝐹))
28	26, 27	jca 516	. . 3 ⊢ ((𝜑 ∧ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))) → ((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)))
29	17, 28	impbida 806	. 2 ⊢ (𝜑 → (((lim inf‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹)) ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))
30	4, 29	bitrd 280	1 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ ((lim sup‘𝐹) ∈ ℝ ∧ (lim sup‘𝐹) ≤ (lim inf‘𝐹))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  dom cdm 5618  ⟶wf 6481  ‘cfv 6485  ℝcr 11028  ℝ^*cxr 11169   ≤ cle 11171  ℤcz 12515  ℤ_≥cuz 12779  lim supclsp 15423   ⇝ cli 15437  lim infclsi 46194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-pm 8766  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9345  df-inf 9346  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-xneg 13054  df-xadd 13055  df-ioo 13293  df-ico 13295  df-fz 13453  df-fzo 13600  df-fl 13742  df-ceil 13743  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15424  df-clim 15441  df-rlim 15442  df-liminf 46195

This theorem is referenced by:  climliminflimsup4  46254