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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim2 | Structured version Visualization version GIF version |
Description: Given a sequence of extended reals, it converges to a real number π΄ w.r.t. the standard topology on the reals (see climreeq 45064), if and only if it converges to π΄ w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim2.m | β’ (π β π β β€) |
xlimclim2.z | β’ π = (β€β₯βπ) |
xlimclim2.f | β’ (π β πΉ:πβΆβ*) |
xlimclim2.a | β’ (π β π΄ β β) |
Ref | Expression |
---|---|
xlimclim2 | β’ (π β (πΉ~~>*π΄ β πΉ β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 β’ ((π β§ πΉ~~>*π΄) β πΉ~~>*π΄) | |
2 | xlimclim2.z | . . . 4 β’ π = (β€β₯βπ) | |
3 | xlimclim2.f | . . . . 5 β’ (π β πΉ:πβΆβ*) | |
4 | 3 | adantr 479 | . . . 4 β’ ((π β§ πΉ~~>*π΄) β πΉ:πβΆβ*) |
5 | xlimclim2.a | . . . . 5 β’ (π β π΄ β β) | |
6 | 5 | adantr 479 | . . . 4 β’ ((π β§ πΉ~~>*π΄) β π΄ β β) |
7 | xlimclim2.m | . . . . . 6 β’ (π β π β β€) | |
8 | 7 | adantr 479 | . . . . 5 β’ ((π β§ πΉ~~>*π΄) β π β β€) |
9 | 8, 2, 4, 6, 1 | xlimxrre 45282 | . . . 4 β’ ((π β§ πΉ~~>*π΄) β βπ β π (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) |
10 | 2, 4, 6, 9 | xlimclim2lem 45290 | . . 3 β’ ((π β§ πΉ~~>*π΄) β (πΉ~~>*π΄ β πΉ β π΄)) |
11 | 1, 10 | mpbid 231 | . 2 β’ ((π β§ πΉ~~>*π΄) β πΉ β π΄) |
12 | simpr 483 | . . 3 β’ ((π β§ πΉ β π΄) β πΉ β π΄) | |
13 | 3 | adantr 479 | . . . 4 β’ ((π β§ πΉ β π΄) β πΉ:πβΆβ*) |
14 | 5 | adantr 479 | . . . 4 β’ ((π β§ πΉ β π΄) β π΄ β β) |
15 | 7 | adantr 479 | . . . . 5 β’ ((π β§ πΉ β π΄) β π β β€) |
16 | 15, 2, 13, 14, 12 | climxrre 45201 | . . . 4 β’ ((π β§ πΉ β π΄) β βπ β π (πΉ βΎ (β€β₯βπ)):(β€β₯βπ)βΆβ) |
17 | 2, 13, 14, 16 | xlimclim2lem 45290 | . . 3 β’ ((π β§ πΉ β π΄) β (πΉ~~>*π΄ β πΉ β π΄)) |
18 | 12, 17 | mpbird 256 | . 2 β’ ((π β§ πΉ β π΄) β πΉ~~>*π΄) |
19 | 11, 18 | impbida 799 | 1 β’ (π β (πΉ~~>*π΄ β πΉ β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5143 βΆwf 6539 βcfv 6543 βcr 11137 β*cxr 11277 β€cz 12588 β€β₯cuz 12852 β cli 15460 ~~>*clsxlim 45269 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fi 9434 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fl 13789 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-starv 17247 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-rest 17403 df-topn 17404 df-topgen 17424 df-ordt 17482 df-ps 18557 df-tsr 18558 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-lm 23151 df-xms 24244 df-ms 24245 df-xlim 45270 |
This theorem is referenced by: climxlim2lem 45296 dfxlim2v 45298 xlimclimdm 45305 |
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