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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 6854 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | limsupvaluz3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 5 | zred 12604 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) | |
8 | 5, 2 | uzinico3 43771 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
9 | 8 | eqcomd 2742 | . . . . 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 | 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 44005 | 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 1541 Ⅎwnf 1785 ∈ wcel 2106 Vcvv 3444 ∩ cin 3908 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7354 +∞cpnf 11183 ℝ*cxr 11185 ℤcz 12496 ℤ≥cuz 12760 -𝑒cxne 13027 [,)cico 13263 lim supclsp 15349 lim infclsi 43962 |
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 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-isom 6503 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-inf 9376 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-n0 12411 df-z 12497 df-uz 12761 df-q 12871 df-xneg 13030 df-ico 13267 df-limsup 15350 df-liminf 43963 |
This theorem is referenced by: limsupvaluz4 44011 |
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