![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lmlimxrge0 | Structured version Visualization version GIF version |
Description: Relate a limit in the nonnegative extended reals to a complex limit, provided the considered function is a real function. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
Ref | Expression |
---|---|
lmlimxrge0.j | β’ π½ = (TopOpenβ(β*π βΎs (0[,]+β))) |
lmlimxrge0.f | β’ (π β πΉ:ββΆπ) |
lmlimxrge0.p | β’ (π β π β π) |
lmlimxrge0.x | β’ π β (0[,)+β) |
Ref | Expression |
---|---|
lmlimxrge0 | β’ (π β (πΉ(βπ‘βπ½)π β πΉ β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmlimxrge0.j | . . . 4 β’ π½ = (TopOpenβ(β*π βΎs (0[,]+β))) | |
2 | xrge0topn 33452 | . . . 4 β’ (TopOpenβ(β*π βΎs (0[,]+β))) = ((ordTopβ β€ ) βΎt (0[,]+β)) | |
3 | 1, 2 | eqtri 2754 | . . 3 β’ π½ = ((ordTopβ β€ ) βΎt (0[,]+β)) |
4 | letopon 23059 | . . . 4 β’ (ordTopβ β€ ) β (TopOnββ*) | |
5 | iccssxr 13410 | . . . 4 β’ (0[,]+β) β β* | |
6 | resttopon 23015 | . . . 4 β’ (((ordTopβ β€ ) β (TopOnββ*) β§ (0[,]+β) β β*) β ((ordTopβ β€ ) βΎt (0[,]+β)) β (TopOnβ(0[,]+β))) | |
7 | 4, 5, 6 | mp2an 689 | . . 3 β’ ((ordTopβ β€ ) βΎt (0[,]+β)) β (TopOnβ(0[,]+β)) |
8 | 3, 7 | eqeltri 2823 | . 2 β’ π½ β (TopOnβ(0[,]+β)) |
9 | lmlimxrge0.f | . 2 β’ (π β πΉ:ββΆπ) | |
10 | lmlimxrge0.p | . 2 β’ (π β π β π) | |
11 | fvex 6897 | . . . 4 β’ (ordTopβ β€ ) β V | |
12 | lmlimxrge0.x | . . . . 5 β’ π β (0[,)+β) | |
13 | icossicc 13416 | . . . . 5 β’ (0[,)+β) β (0[,]+β) | |
14 | 12, 13 | sstri 3986 | . . . 4 β’ π β (0[,]+β) |
15 | ovex 7437 | . . . 4 β’ (0[,]+β) β V | |
16 | restabs 23019 | . . . 4 β’ (((ordTopβ β€ ) β V β§ π β (0[,]+β) β§ (0[,]+β) β V) β (((ordTopβ β€ ) βΎt (0[,]+β)) βΎt π) = ((ordTopβ β€ ) βΎt π)) | |
17 | 11, 14, 15, 16 | mp3an 1457 | . . 3 β’ (((ordTopβ β€ ) βΎt (0[,]+β)) βΎt π) = ((ordTopβ β€ ) βΎt π) |
18 | 3 | oveq1i 7414 | . . 3 β’ (π½ βΎt π) = (((ordTopβ β€ ) βΎt (0[,]+β)) βΎt π) |
19 | rge0ssre 13436 | . . . . 5 β’ (0[,)+β) β β | |
20 | 12, 19 | sstri 3986 | . . . 4 β’ π β β |
21 | eqid 2726 | . . . . 5 β’ (TopOpenββfld) = (TopOpenββfld) | |
22 | eqid 2726 | . . . . 5 β’ (ordTopβ β€ ) = (ordTopβ β€ ) | |
23 | 21, 22 | xrrest2 24674 | . . . 4 β’ (π β β β ((TopOpenββfld) βΎt π) = ((ordTopβ β€ ) βΎt π)) |
24 | 20, 23 | ax-mp 5 | . . 3 β’ ((TopOpenββfld) βΎt π) = ((ordTopβ β€ ) βΎt π) |
25 | 17, 18, 24 | 3eqtr4i 2764 | . 2 β’ (π½ βΎt π) = ((TopOpenββfld) βΎt π) |
26 | ax-resscn 11166 | . . 3 β’ β β β | |
27 | 20, 26 | sstri 3986 | . 2 β’ π β β |
28 | 8, 9, 10, 25, 27 | lmlim 33456 | 1 β’ (π β (πΉ(βπ‘βπ½)π β πΉ β π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 class class class wbr 5141 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 0cc0 11109 +βcpnf 11246 β*cxr 11248 β€ cle 11250 βcn 12213 [,)cico 13329 [,]cicc 13330 β cli 15431 βΎs cress 17179 βΎt crest 17372 TopOpenctopn 17373 ordTopcordt 17451 β*π cxrs 17452 βfldccnfld 21235 TopOnctopon 22762 βπ‘clm 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-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 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-fi 9405 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-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-rest 17374 df-topn 17375 df-topgen 17395 df-ordt 17453 df-xrs 17454 df-ps 18528 df-tsr 18529 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-lm 23083 df-xms 24176 df-ms 24177 |
This theorem is referenced by: esumcvg 33613 dstfrvclim1 34005 |
Copyright terms: Public domain | W3C validator |