![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfvaluz3 | 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 |
---|---|
liminfvaluz3.1 | ⊢ Ⅎ𝑘𝜑 |
liminfvaluz3.2 | ⊢ Ⅎ𝑘𝐹 |
liminfvaluz3.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
liminfvaluz3.4 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
liminfvaluz3.5 | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
Ref | Expression |
---|---|
liminfvaluz3 | ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2926 | . . . 4 ⊢ Ⅎ𝑘𝑍 | |
2 | liminfvaluz3.2 | . . . 4 ⊢ Ⅎ𝑘𝐹 | |
3 | liminfvaluz3.5 | . . . 4 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 1, 2, 3 | feqmptdf 6558 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) |
5 | 4 | fveq2d 6497 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) = (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘)))) |
6 | liminfvaluz3.1 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
7 | liminfvaluz3.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | liminfvaluz3.4 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
9 | 3 | ffvelrnda 6670 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ*) |
10 | 6, 7, 8, 9 | liminfvaluz 41450 | . 2 ⊢ (𝜑 → (lim inf‘(𝑘 ∈ 𝑍 ↦ (𝐹‘𝑘))) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)))) |
11 | 5, 10 | eqtrd 2808 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = -𝑒(lim sup‘(𝑘 ∈ 𝑍 ↦ -𝑒(𝐹‘𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2048 Ⅎwnfc 2910 ↦ cmpt 5002 ⟶wf 6178 ‘cfv 6182 ℝ*cxr 10465 ℤcz 11786 ℤ≥cuz 12051 -𝑒cxne 12314 lim supclsp 14678 lim infclsi 41409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-xneg 12317 df-ico 12553 df-limsup 14679 df-liminf 41410 |
This theorem is referenced by: liminflbuz2 41473 liminfpnfuz 41474 |
Copyright terms: Public domain | W3C validator |