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 6683 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | limsupvaluz3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 5 | zred 12086 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) | |
8 | 5, 2 | uzinico3 41837 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
9 | 8 | eqcomd 2827 | . . . . 5 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
11 | 7, 10 | eleqtrd 2915 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ 𝑍) |
12 | limsupvaluz3.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
13 | 11, 12 | syldan 593 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
14 | 1, 4, 6, 13 | limsupval4 42073 | 1 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 +∞cpnf 10671 ℝ*cxr 10673 ℤcz 11980 ℤ≥cuz 12242 -𝑒cxne 12503 [,)cico 12739 lim supclsp 14826 lim infclsi 42030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-xneg 12506 df-ico 12743 df-limsup 14827 df-liminf 42031 |
This theorem is referenced by: limsupvaluz4 42079 |
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