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Mirrors > Home > MPE Home > Th. List > rlimcnp3 | Structured version Visualization version GIF version |
Description: Relate a limit of a real-valued sequence at infinity to the continuity of the function π(π¦) = π (1 / π¦) at zero. (Contributed by Mario Carneiro, 1-Mar-2015.) |
Ref | Expression |
---|---|
rlimcnp3.c | β’ (π β πΆ β β) |
rlimcnp3.r | β’ ((π β§ π¦ β β+) β π β β) |
rlimcnp3.s | β’ (π¦ = (1 / π₯) β π = π ) |
rlimcnp3.j | β’ π½ = (TopOpenββfld) |
rlimcnp3.k | β’ πΎ = (π½ βΎt (0[,)+β)) |
Ref | Expression |
---|---|
rlimcnp3 | β’ (π β ((π¦ β β+ β¦ π) βπ πΆ β (π₯ β (0[,)+β) β¦ if(π₯ = 0, πΆ, π )) β ((πΎ CnP π½)β0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3971 | . 2 β’ (π β (0[,)+β) β (0[,)+β)) | |
2 | 0e0icopnf 13384 | . . 3 β’ 0 β (0[,)+β) | |
3 | 2 | a1i 11 | . 2 β’ (π β 0 β (0[,)+β)) |
4 | rpssre 12930 | . . 3 β’ β+ β β | |
5 | 4 | a1i 11 | . 2 β’ (π β β+ β β) |
6 | rlimcnp3.c | . 2 β’ (π β πΆ β β) | |
7 | rlimcnp3.r | . 2 β’ ((π β§ π¦ β β+) β π β β) | |
8 | simpr 486 | . . 3 β’ ((π β§ π¦ β β+) β π¦ β β+) | |
9 | rpreccl 12949 | . . . . . 6 β’ (π¦ β β+ β (1 / π¦) β β+) | |
10 | 9 | adantl 483 | . . . . 5 β’ ((π β§ π¦ β β+) β (1 / π¦) β β+) |
11 | 10 | rpred 12965 | . . . 4 β’ ((π β§ π¦ β β+) β (1 / π¦) β β) |
12 | 10 | rpge0d 12969 | . . . 4 β’ ((π β§ π¦ β β+) β 0 β€ (1 / π¦)) |
13 | elrege0 13380 | . . . 4 β’ ((1 / π¦) β (0[,)+β) β ((1 / π¦) β β β§ 0 β€ (1 / π¦))) | |
14 | 11, 12, 13 | sylanbrc 584 | . . 3 β’ ((π β§ π¦ β β+) β (1 / π¦) β (0[,)+β)) |
15 | 8, 14 | 2thd 265 | . 2 β’ ((π β§ π¦ β β+) β (π¦ β β+ β (1 / π¦) β (0[,)+β))) |
16 | rlimcnp3.s | . 2 β’ (π¦ = (1 / π₯) β π = π ) | |
17 | rlimcnp3.j | . 2 β’ π½ = (TopOpenββfld) | |
18 | rlimcnp3.k | . 2 β’ πΎ = (π½ βΎt (0[,)+β)) | |
19 | 1, 3, 5, 6, 7, 15, 16, 17, 18 | rlimcnp2 26339 | 1 β’ (π β ((π¦ β β+ β¦ π) βπ πΆ β (π₯ β (0[,)+β) β¦ if(π₯ = 0, πΆ, π )) β ((πΎ CnP π½)β0))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3914 ifcif 4490 class class class wbr 5109 β¦ cmpt 5192 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 0cc0 11059 1c1 11060 +βcpnf 11194 β€ cle 11198 / cdiv 11820 β+crp 12923 [,)cico 13275 βπ crli 15376 βΎt crest 17310 TopOpenctopn 17311 βfldccnfld 20819 CnP ccnp 22599 |
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-tp 4595 df-op 4597 df-uni 4870 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-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 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-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ico 13279 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-rlim 15380 df-struct 17027 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-starv 17156 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-rest 17312 df-topn 17313 df-topgen 17333 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-cnfld 20820 df-top 22266 df-topon 22283 df-bases 22319 df-cnp 22602 |
This theorem is referenced by: efrlim 26342 |
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