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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 6915 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
5 | limsupvaluz3.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | 5 | zred 12718 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
7 | simpr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) | |
8 | 5, 2 | uzinico3 45181 | . . . . . 6 ⊢ (𝜑 → 𝑍 = (𝑍 ∩ (𝑀[,)+∞))) |
9 | 8 | eqcomd 2732 | . . . . 5 ⊢ (𝜑 → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
10 | 9 | adantr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → (𝑍 ∩ (𝑀[,)+∞)) = 𝑍) |
11 | 7, 10 | eleqtrd 2828 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝑘 ∈ 𝑍) |
12 | limsupvaluz3.b | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
13 | 11, 12 | syldan 589 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑍 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
14 | 1, 4, 6, 13 | limsupval4 45415 | 1 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝑍 ↦ 𝐵)) = -𝑒(lim inf‘(𝑘 ∈ 𝑍 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Vcvv 3462 ∩ cin 3946 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 +∞cpnf 11295 ℝ*cxr 11297 ℤcz 12610 ℤ≥cuz 12874 -𝑒cxne 13143 [,)cico 13380 lim supclsp 15472 lim infclsi 45372 |
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 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-inf 9486 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-q 12985 df-xneg 13146 df-ico 13384 df-limsup 15473 df-liminf 45373 |
This theorem is referenced by: limsupvaluz4 45421 |
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