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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfvaluz | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf for an extended real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfvaluz.k | ⊢ Ⅎ𝑘𝜑 |
liminfvaluz.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminfvaluz.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminfvaluz.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
liminfvaluz | ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminfvaluz.k | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | liminfvaluz.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | fvexi 6853 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | liminfvaluz.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 5 | zred 12565 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) | |
8 | 5, 2 | uzinico3 43695 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
9 | 8 | eqcomd 2743 | . . . . 5 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
11 | 7, 10 | eleqtrd 2840 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ 𝑍) |
12 | liminfvaluz.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
13 | 11, 12 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
14 | 1, 4, 6, 13 | liminfval3 43925 | 1 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Vcvv 3443 ∩ cin 3907 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 +∞cpnf 11144 ℝ*cxr 11146 ℤcz 12457 ℤ≥cuz 12721 -𝑒cxne 12984 [,)cico 13220 lim supclsp 15311 lim infclsi 43886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-xneg 12987 df-ico 13224 df-limsup 15312 df-liminf 43887 |
This theorem is referenced by: liminfvaluz2 43930 liminfvaluz3 43931 |
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