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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 6961 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
3 | 2 | fveq2d 6895 | . . . . 5 ⊢ (𝜑 → (lim inf‘𝐹) = (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))) |
4 | climliminflimsupd.2 | . . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 4 | fvexi 6905 | . . . . . . . 8 ⊢ 𝑍 ∈ V |
6 | 5 | mptex 7229 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)) ∈ V |
7 | liminfcl 45074 | . . . . . . 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 2828 | . . . 4 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ*) |
11 | nfv 1910 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
12 | climliminflimsupd.1 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
13 | 1 | ffvelcdmda 7088 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | 13 | renegcld 11663 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → -(𝐹‘𝑘) ∈ ℝ) |
15 | 11, 12, 4, 14 | limsupvaluz4 45111 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)))) |
16 | climrel 15460 | . . . . . . . . . 10 ⊢ Rel ⇝ | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → Rel ⇝ ) |
18 | nfcv 2898 | . . . . . . . . . 10 ⊢ Ⅎ𝑘𝐹 | |
19 | climliminflimsupd.4 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
20 | 12, 4, 1 | climlimsup 45071 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ (lim sup‘𝐹))) |
21 | 19, 20 | mpbid 231 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⇝ (lim sup‘𝐹)) |
22 | 13 | recnd 11264 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
23 | 11, 18, 4, 12, 21, 22 | climneg 44921 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) |
24 | releldm 5940 | . . . . . . . . 9 ⊢ ((Rel ⇝ ∧ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ -(lim sup‘𝐹)) → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) | |
25 | 17, 23, 24 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ) |
26 | 14 | fmpttd 7119 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)):𝑍⟶ℝ) |
27 | 12, 4, 26 | climlimsup 45071 | . . . . . . . 8 ⊢ (𝜑 → ((𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ∈ dom ⇝ ↔ (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))))) |
28 | 25, 27 | mpbid 231 | . . . . . . 7 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)) ⇝ (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘)))) |
29 | climuni 15520 | . . . . . . 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 583 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ -(𝐹‘𝑘))) = -(lim sup‘𝐹)) |
31 | 22 | negnegd 11584 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → --(𝐹‘𝑘) = (𝐹‘𝑘)) |
32 | 31 | mpteq2dva 5242 | . . . . . . . . 9 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
33 | 32, 2 | eqtr4d 2770 | . . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘)) = 𝐹) |
34 | 33 | fveq2d 6895 | . . . . . . 7 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = (lim inf‘𝐹)) |
35 | 34 | xnegeqd 44742 | . . . . . 6 ⊢ (𝜑 → -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ --(𝐹‘𝑘))) = -𝑒(lim inf‘𝐹)) |
36 | 15, 30, 35 | 3eqtr3d 2775 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) = -𝑒(lim inf‘𝐹)) |
37 | 4, 12, 21, 13 | climrecl 15551 | . . . . . 6 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℝ) |
38 | 37 | renegcld 11663 | . . . . 5 ⊢ (𝜑 → -(lim sup‘𝐹) ∈ ℝ) |
39 | 36, 38 | eqeltrrd 2829 | . . . 4 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) ∈ ℝ) |
40 | xnegrecl2 44765 | . . . 4 ⊢ (((lim inf‘𝐹) ∈ ℝ* ∧ -𝑒(lim inf‘𝐹) ∈ ℝ) → (lim inf‘𝐹) ∈ ℝ) | |
41 | 10, 39, 40 | syl2anc 583 | . . 3 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℝ) |
42 | 41 | recnd 11264 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) ∈ ℂ) |
43 | 37 | recnd 11264 | . 2 ⊢ (𝜑 → (lim sup‘𝐹) ∈ ℂ) |
44 | 41 | rexnegd 44432 | . . 3 ⊢ (𝜑 → -𝑒(lim inf‘𝐹) = -(lim inf‘𝐹)) |
45 | 36, 44 | eqtr2d 2768 | . 2 ⊢ (𝜑 → -(lim inf‘𝐹) = -(lim sup‘𝐹)) |
46 | 42, 43, 45 | neg11d 11605 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = (lim sup‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5672 Rel wrel 5677 ⟶wf 6538 ‘cfv 6542 ℝcr 11129 ℝ*cxr 11269 -cneg 11467 ℤcz 12580 ℤ≥cuz 12844 -𝑒cxne 13113 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-er 8718 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 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-ico 13354 df-fl 13781 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: climliminf 45117 climliminflimsup 45119 climliminflimsup2 45120 xlimliminflimsup 45173 |
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