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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 4000 | . 2 β’ (π β (0[,)+β) β (0[,)+β)) | |
2 | 0e0icopnf 13438 | . . 3 β’ 0 β (0[,)+β) | |
3 | 2 | a1i 11 | . 2 β’ (π β 0 β (0[,)+β)) |
4 | rpssre 12984 | . . 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 484 | . . 3 β’ ((π β§ π¦ β β+) β π¦ β β+) | |
9 | rpreccl 13003 | . . . . . 6 β’ (π¦ β β+ β (1 / π¦) β β+) | |
10 | 9 | adantl 481 | . . . . 5 β’ ((π β§ π¦ β β+) β (1 / π¦) β β+) |
11 | 10 | rpred 13019 | . . . 4 β’ ((π β§ π¦ β β+) β (1 / π¦) β β) |
12 | 10 | rpge0d 13023 | . . . 4 β’ ((π β§ π¦ β β+) β 0 β€ (1 / π¦)) |
13 | elrege0 13434 | . . . 4 β’ ((1 / π¦) β (0[,)+β) β ((1 / π¦) β β β§ 0 β€ (1 / π¦))) | |
14 | 11, 12, 13 | sylanbrc 582 | . . 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 26849 | 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 395 = wceq 1533 β wcel 2098 β wss 3943 ifcif 4523 class class class wbr 5141 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 0cc0 11109 1c1 11110 +βcpnf 11246 β€ cle 11250 / cdiv 11872 β+crp 12977 [,)cico 13329 βπ crli 15433 βΎt crest 17373 TopOpenctopn 17374 βfldccnfld 21236 CnP ccnp 23080 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-fz 13488 df-seq 13970 df-exp 14031 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-rlim 15437 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-rest 17375 df-topn 17376 df-topgen 17396 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-cnfld 21237 df-top 22747 df-topon 22764 df-bases 22800 df-cnp 23083 |
This theorem is referenced by: efrlim 26852 efrlimOLD 26853 |
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