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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climliminflimsupd | Structured version Visualization version GIF version |
Description: If a sequence of real numbers converges, its inferior limit and its superior limit are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
climliminflimsupd.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climliminflimsupd.2 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climliminflimsupd.3 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
climliminflimsupd.4 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
Ref | Expression |
---|---|
climliminflimsupd | ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climliminflimsupd.3 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
2 | 1 | feqmptd 6496 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
3 | 2 | fveq2d 6437 | . . . . 5 ⊢ (𝜑 → (lim inf‘𝐹) = (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))) |
4 | climliminflimsupd.2 | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | fvexi 6447 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 6742 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | liminfcl 40790 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V → (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ*) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ* |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) ∈ ℝ*) |
10 | 3, 9 | eqeltrd 2906 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) |
11 | nfv 2015 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
12 | climliminflimsupd.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 1 | ffvelrnda 6608 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | 13 | renegcld 10781 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
15 | 11, 12, 4, 14 | limsupvaluz4 40827 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)))) |
16 | climrel 14600 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel ⇝ ) |
18 | nfcv 2969 | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝐹 | |
19 | climliminflimsupd.4 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
20 | 12, 4, 1 | climlimsup 40787 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
21 | 19, 20 | mpbid 224 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
22 | 13 | recnd 10385 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
23 | 11, 18, 4, 12, 21, 22 | climneg 40637 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) |
24 | releldm 5591 | . . . . . . . . 9 ⊢ ((Rel ⇝ ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) | |
25 | 17, 23, 24 | syl2anc 581 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) |
26 | 14 | fmpttd 6634 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)):𝑍⟶ℝ) |
27 | 12, 4, 26 | climlimsup 40787 | . . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ↔ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))))) |
28 | 25, 27 | mpbid 224 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) |
29 | climuni 14660 | . . . . . . 7 ⊢ (((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) | |
30 | 28, 23, 29 | syl2anc 581 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) |
31 | 22 | negnegd 10704 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → --(𝐹‘𝑘) = (𝐹‘𝑘)) |
32 | 31 | mpteq2dva 4967 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
33 | 32, 2 | eqtr4d 2864 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = 𝐹) |
34 | 33 | fveq2d 6437 | . . . . . . 7 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = (lim inf‘𝐹)) |
35 | 34 | xnegeqd 40459 | . . . . . 6 ⊢ (𝜑 → -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = -𝑒(lim inf‘𝐹)) |
36 | 15, 30, 35 | 3eqtr3d 2869 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) = -𝑒(lim inf‘𝐹)) |
37 | 4, 12, 21, 13 | climrecl 14691 | . . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
38 | 37 | renegcld 10781 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) ∈ ℝ) |
39 | 36, 38 | eqeltrrd 2907 | . . . 4 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) ∈ ℝ) |
40 | xnegrecl2 40484 | . . . 4 ⊢ (((lim inf‘𝐹) ∈ ℝ* ∧ -𝑒(lim inf‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ) | |
41 | 10, 39, 40 | syl2anc 581 | . . 3 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) |
42 | 41 | recnd 10385 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℂ) |
43 | 37 | recnd 10385 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℂ) |
44 | 41 | rexnegd 40144 | . . 3 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) = -(lim inf‘𝐹)) |
45 | 36, 44 | eqtr2d 2862 | . 2 ⊢ (𝜑 → -(lim inf‘𝐹) = -(lim sup‘𝐹)) |
46 | 42, 43, 45 | neg11d 10725 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 ↦ cmpt 4952 dom cdm 5342 Rel wrel 5347 ⟶wf 6119 ‘cfv 6123 ℝcr 10251 ℝ*cxr 10390 -cneg 10586 ℤcz 11704 ℤ≥cuz 11968 -𝑒cxne 12229 lim supclsp 14578 ⇝ cli 14592 lim infclsi 40778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-n0 11619 df-z 11705 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-ico 12469 df-fl 12888 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-limsup 14579 df-clim 14596 df-rlim 14597 df-liminf 40779 |
This theorem is referenced by: climliminf 40833 climliminflimsup 40835 climliminflimsup2 40836 |
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