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 6726 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
3 | 2 | fveq2d 6667 | . . . . 5 ⊢ (𝜑 → (lim inf‘𝐹) = (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))) |
4 | climliminflimsupd.2 | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | fvexi 6677 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 6978 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | liminfcl 42034 | . . . . . . 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 2911 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) |
11 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
12 | climliminflimsupd.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 1 | ffvelrnda 6844 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | 13 | renegcld 11059 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
15 | 11, 12, 4, 14 | limsupvaluz4 42071 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)))) |
16 | climrel 14841 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel ⇝ ) |
18 | nfcv 2975 | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝐹 | |
19 | climliminflimsupd.4 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
20 | 12, 4, 1 | climlimsup 42031 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
21 | 19, 20 | mpbid 234 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
22 | 13 | recnd 10661 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
23 | 11, 18, 4, 12, 21, 22 | climneg 41881 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) |
24 | releldm 5807 | . . . . . . . . 9 ⊢ ((Rel ⇝ ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) | |
25 | 17, 23, 24 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) |
26 | 14 | fmpttd 6872 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)):𝑍⟶ℝ) |
27 | 12, 4, 26 | climlimsup 42031 | . . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ↔ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))))) |
28 | 25, 27 | mpbid 234 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) |
29 | climuni 14901 | . . . . . . 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 586 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) |
31 | 22 | negnegd 10980 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → --(𝐹‘𝑘) = (𝐹‘𝑘)) |
32 | 31 | mpteq2dva 5152 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
33 | 32, 2 | eqtr4d 2857 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = 𝐹) |
34 | 33 | fveq2d 6667 | . . . . . . 7 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = (lim inf‘𝐹)) |
35 | 34 | xnegeqd 41701 | . . . . . 6 ⊢ (𝜑 → -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = -𝑒(lim inf‘𝐹)) |
36 | 15, 30, 35 | 3eqtr3d 2862 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) = -𝑒(lim inf‘𝐹)) |
37 | 4, 12, 21, 13 | climrecl 14932 | . . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
38 | 37 | renegcld 11059 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) ∈ ℝ) |
39 | 36, 38 | eqeltrrd 2912 | . . . 4 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) ∈ ℝ) |
40 | xnegrecl2 41726 | . . . 4 ⊢ (((lim inf‘𝐹) ∈ ℝ* ∧ -𝑒(lim inf‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ) | |
41 | 10, 39, 40 | syl2anc 586 | . . 3 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) |
42 | 41 | recnd 10661 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℂ) |
43 | 37 | recnd 10661 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℂ) |
44 | 41 | rexnegd 41402 | . . 3 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) = -(lim inf‘𝐹)) |
45 | 36, 44 | eqtr2d 2855 | . 2 ⊢ (𝜑 → -(lim inf‘𝐹) = -(lim sup‘𝐹)) |
46 | 42, 43, 45 | neg11d 11001 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 Vcvv 3493 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 Rel wrel 5553 ⟶wf 6344 ‘cfv 6348 ℝcr 10528 ℝ*cxr 10666 -cneg 10863 ℤcz 11973 ℤ≥cuz 12235 -𝑒cxne 12496 lim supclsp 14819 ⇝ cli 14833 lim infclsi 42022 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-ico 12736 df-fl 13154 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-liminf 42023 |
This theorem is referenced by: climliminf 42077 climliminflimsup 42079 climliminflimsup2 42080 xlimliminflimsup 42133 |
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