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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimmnflimsup2 | Structured version Visualization version GIF version |
Description: A sequence of extended reals converges to -β if and only if its superior limit is also -β. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
xlimmnflimsup2.m | β’ (π β π β β€) |
xlimmnflimsup2.z | β’ π = (β€β₯βπ) |
xlimmnflimsup2.f | β’ (π β πΉ:πβΆβ*) |
Ref | Expression |
---|---|
xlimmnflimsup2 | β’ (π β (πΉ~~>*-β β (lim supβπΉ) = -β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimmnflimsup2.m | . . 3 β’ (π β π β β€) | |
2 | xlimmnflimsup2.z | . . 3 β’ π = (β€β₯βπ) | |
3 | xlimmnflimsup2.f | . . 3 β’ (π β πΉ:πβΆβ*) | |
4 | 1, 2, 3 | xlimmnfv 44165 | . 2 β’ (π β (πΉ~~>*-β β βπ₯ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
5 | nfcv 2904 | . . 3 β’ β²ππΉ | |
6 | 5, 1, 2, 3 | limsupmnfuz 44058 | . 2 β’ (π β ((lim supβπΉ) = -β β βπ₯ β β βπ β π βπ β (β€β₯βπ)(πΉβπ) β€ π₯)) |
7 | 4, 6 | bitr4d 282 | 1 β’ (π β (πΉ~~>*-β β (lim supβπΉ) = -β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 βwral 3061 βwrex 3070 class class class wbr 5109 βΆwf 6496 βcfv 6500 βcr 11058 -βcmnf 11195 β*cxr 11196 β€ cle 11198 β€cz 12507 β€β₯cuz 12771 lim supclsp 15361 ~~>*clsxlim 44149 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-n0 12422 df-z 12508 df-uz 12772 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fl 13706 df-ceil 13707 df-limsup 15362 df-topgen 17333 df-ordt 17391 df-ps 18463 df-tsr 18464 df-top 22266 df-topon 22283 df-bases 22319 df-lm 22603 df-xlim 44150 |
This theorem is referenced by: xlimliminflimsup 44193 |
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