Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupvaluz3 | 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 |
---|---|
limsupvaluz3.k | ⊢ Ⅎ𝑘𝜑 |
limsupvaluz3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupvaluz3.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
limsupvaluz3.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
limsupvaluz3 | ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupvaluz3.k | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | limsupvaluz3.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | 2 | fvexi 6677 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | limsupvaluz3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 5 | zred 12075 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) | |
8 | 5, 2 | uzinico3 41715 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
9 | 8 | eqcomd 2824 | . . . . 5 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
11 | 7, 10 | eleqtrd 2912 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ 𝑍) |
12 | limsupvaluz3.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
13 | 11, 12 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
14 | 1, 4, 6, 13 | limsupval4 41951 | 1 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 +∞cpnf 10660 ℝ*cxr 10662 ℤcz 11969 ℤ≥cuz 12231 -𝑒cxne 12492 [,)cico 12728 lim supclsp 14815 lim infclsi 41908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-xneg 12495 df-ico 12732 df-limsup 14816 df-liminf 41909 |
This theorem is referenced by: limsupvaluz4 41957 |
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