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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmclimxlim | Structured version Visualization version GIF version |
Description: A real valued sequence that converges w.r.t. the topology on the complex numbers, converges w.r.t. the topology on the extended reals (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
Ref | Expression |
---|---|
dmclimxlim.1 | β’ (π β π β β€) |
dmclimxlim.2 | β’ π = (β€β₯βπ) |
dmclimxlim.3 | β’ (π β πΉ:πβΆβ) |
dmclimxlim.4 | β’ (π β πΉ β dom β ) |
Ref | Expression |
---|---|
dmclimxlim | β’ (π β πΉ β dom ~~>*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimrel 43816 | . 2 β’ Rel ~~>* | |
2 | dmclimxlim.1 | . . 3 β’ (π β π β β€) | |
3 | dmclimxlim.2 | . . 3 β’ π = (β€β₯βπ) | |
4 | dmclimxlim.3 | . . 3 β’ (π β πΉ:πβΆβ) | |
5 | dmclimxlim.4 | . . . 4 β’ (π β πΉ β dom β ) | |
6 | 2, 3, 4 | climliminf 43802 | . . . 4 β’ (π β (πΉ β dom β β πΉ β (lim infβπΉ))) |
7 | 5, 6 | mpbid 231 | . . 3 β’ (π β πΉ β (lim infβπΉ)) |
8 | 2, 3, 4, 7 | climxlim 43822 | . 2 β’ (π β πΉ~~>*(lim infβπΉ)) |
9 | releldm 5896 | . 2 β’ ((Rel ~~>* β§ πΉ~~>*(lim infβπΉ)) β πΉ β dom ~~>*) | |
10 | 1, 8, 9 | sylancr 588 | 1 β’ (π β πΉ β dom ~~>*) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 class class class wbr 5104 dom cdm 5631 Rel wrel 5636 βΆwf 6488 βcfv 6492 βcr 10984 β€cz 12433 β€β₯cuz 12696 β cli 15301 lim infclsi 43747 ~~>*clsxlim 43814 |
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 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-er 8582 df-map 8701 df-pm 8702 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fi 9281 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-q 12803 df-rp 12845 df-xneg 12962 df-xadd 12963 df-xmul 12964 df-ioo 13197 df-ioc 13198 df-ico 13199 df-icc 13200 df-fz 13354 df-fl 13626 df-seq 13836 df-exp 13897 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-limsup 15288 df-clim 15305 df-rlim 15306 df-struct 16954 df-slot 16989 df-ndx 17001 df-base 17019 df-plusg 17081 df-mulr 17082 df-starv 17083 df-tset 17087 df-ple 17088 df-ds 17090 df-unif 17091 df-rest 17239 df-topn 17240 df-topgen 17260 df-ordt 17318 df-ps 18390 df-tsr 18391 df-psmet 20712 df-xmet 20713 df-met 20714 df-bl 20715 df-mopn 20716 df-cnfld 20721 df-top 22166 df-topon 22183 df-topsp 22205 df-bases 22219 df-lm 22503 df-xms 23596 df-ms 23597 df-liminf 43748 df-xlim 43815 |
This theorem is referenced by: xlimliminflimsup 43858 |
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