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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfvaluz4 | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf for a real-valued function, defined on a set of upper integers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfvaluz4.1 | β’ β²ππ |
liminfvaluz4.2 | β’ β²ππΉ |
liminfvaluz4.3 | β’ (π β π β β€) |
liminfvaluz4.4 | β’ π = (β€β₯βπ) |
liminfvaluz4.5 | β’ (π β πΉ:πβΆβ) |
Ref | Expression |
---|---|
liminfvaluz4 | β’ (π β (lim infβπΉ) = -π(lim supβ(π β π β¦ -(πΉβπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2904 | . . . 4 β’ β²ππ | |
2 | liminfvaluz4.2 | . . . 4 β’ β²ππΉ | |
3 | liminfvaluz4.5 | . . . 4 β’ (π β πΉ:πβΆβ) | |
4 | 1, 2, 3 | feqmptdf 6963 | . . 3 β’ (π β πΉ = (π β π β¦ (πΉβπ))) |
5 | 4 | fveq2d 6896 | . 2 β’ (π β (lim infβπΉ) = (lim infβ(π β π β¦ (πΉβπ)))) |
6 | liminfvaluz4.1 | . . 3 β’ β²ππ | |
7 | liminfvaluz4.3 | . . 3 β’ (π β π β β€) | |
8 | liminfvaluz4.4 | . . 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 | liminfvaluz2 44511 | . 2 β’ (π β (lim infβ(π β π β¦ (πΉβπ))) = -π(lim supβ(π β π β¦ -(πΉβπ)))) |
11 | 5, 10 | eqtrd 2773 | 1 β’ (π β (lim infβπΉ) = -π(lim supβ(π β π β¦ -(πΉβπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β²wnf 1786 β wcel 2107 β²wnfc 2884 β¦ cmpt 5232 βΆwf 6540 βcfv 6544 βcr 11109 -cneg 11445 β€cz 12558 β€β₯cuz 12822 -πcxne 13089 lim supclsp 15414 lim infclsi 44467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-xneg 13092 df-ico 13330 df-limsup 15415 df-liminf 44468 |
This theorem is referenced by: liminfreuzlem 44518 liminfltlem 44520 |
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