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Mathbox for Glauco Siliprandi |
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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 | β’ β²ππ |
liminfvaluz3.2 | β’ β²ππΉ |
liminfvaluz3.3 | β’ (π β π β β€) |
liminfvaluz3.4 | β’ π = (β€β₯βπ) |
liminfvaluz3.5 | β’ (π β πΉ:πβΆβ*) |
Ref | Expression |
---|---|
liminfvaluz3 | β’ (π β (lim infβπΉ) = -π(lim supβ(π β π β¦ -π(πΉβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2892 | . . . 4 β’ β²ππ | |
2 | liminfvaluz3.2 | . . . 4 β’ β²ππΉ | |
3 | liminfvaluz3.5 | . . . 4 β’ (π β πΉ:πβΆβ*) | |
4 | 1, 2, 3 | feqmptdf 6962 | . . 3 β’ (π β πΉ = (π β π β¦ (πΉβπ))) |
5 | 4 | fveq2d 6894 | . 2 β’ (π β (lim infβπΉ) = (lim infβ(π β π β¦ (πΉβπ)))) |
6 | liminfvaluz3.1 | . . 3 β’ β²ππ | |
7 | liminfvaluz3.3 | . . 3 β’ (π β π β β€) | |
8 | liminfvaluz3.4 | . . 3 β’ π = (β€β₯βπ) | |
9 | 3 | ffvelcdmda 7087 | . . 3 β’ ((π β§ π β π) β (πΉβπ) β β*) |
10 | 6, 7, 8, 9 | liminfvaluz 45239 | . 2 β’ (π β (lim infβ(π β π β¦ (πΉβπ))) = -π(lim supβ(π β π β¦ -π(πΉβπ)))) |
11 | 5, 10 | eqtrd 2765 | 1 β’ (π β (lim infβπΉ) = -π(lim supβ(π β π β¦ -π(πΉβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β²wnf 1777 β wcel 2098 β²wnfc 2875 β¦ cmpt 5227 βΆwf 6539 βcfv 6543 β*cxr 11272 β€cz 12583 β€β₯cuz 12847 -πcxne 13116 lim supclsp 15441 lim infclsi 45198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-sup 9460 df-inf 9461 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-q 12958 df-xneg 13119 df-ico 13357 df-limsup 15442 df-liminf 45199 |
This theorem is referenced by: liminflbuz2 45262 liminfpnfuz 45263 |
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