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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 44901), 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 484 | . . 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 480 | . . . 4 β’ ((π β§ πΉ~~>*π΄) β πΉ:πβΆβ*) |
5 | xlimclim2.a | . . . . 5 β’ (π β π΄ β β) | |
6 | 5 | adantr 480 | . . . 4 β’ ((π β§ πΉ~~>*π΄) β π΄ β β) |
7 | xlimclim2.m | . . . . . 6 β’ (π β π β β€) | |
8 | 7 | adantr 480 | . . . . 5 β’ ((π β§ πΉ~~>*π΄) β π β β€) |
9 | 8, 2, 4, 6, 1 | xlimxrre 45119 | . . . 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 45127 | . . 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 484 | . . 3 β’ ((π β§ πΉ β π΄) β πΉ β π΄) | |
13 | 3 | adantr 480 | . . . 4 β’ ((π β§ πΉ β π΄) β πΉ:πβΆβ*) |
14 | 5 | adantr 480 | . . . 4 β’ ((π β§ πΉ β π΄) β π΄ β β) |
15 | 7 | adantr 480 | . . . . 5 β’ ((π β§ πΉ β π΄) β π β β€) |
16 | 15, 2, 13, 14, 12 | climxrre 45038 | . . . 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 45127 | . . 3 β’ ((π β§ πΉ β π΄) β (πΉ~~>*π΄ β πΉ β π΄)) |
18 | 12, 17 | mpbird 257 | . 2 β’ ((π β§ πΉ β π΄) β πΉ~~>*π΄) |
19 | 11, 18 | impbida 798 | 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 395 = wceq 1533 β wcel 2098 class class class wbr 5141 βΆwf 6533 βcfv 6537 βcr 11111 β*cxr 11251 β€cz 12562 β€β₯cuz 12826 β cli 15434 ~~>*clsxlim 45106 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fl 13763 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-rest 17377 df-topn 17378 df-topgen 17398 df-ordt 17456 df-ps 18531 df-tsr 18532 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-lm 23088 df-xms 24181 df-ms 24182 df-xlim 45107 |
This theorem is referenced by: climxlim2lem 45133 dfxlim2v 45135 xlimclimdm 45142 |
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