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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmlim | Structured version Visualization version GIF version |
Description: Relate a limit in a given topology to a complex number limit, provided that topology agrees with the common topology on β on the required subset. (Contributed by Thierry Arnoux, 11-Jul-2017.) |
Ref | Expression |
---|---|
lmlim.j | β’ π½ β (TopOnβπ) |
lmlim.f | β’ (π β πΉ:ββΆπ) |
lmlim.p | β’ (π β π β π) |
lmlim.t | β’ (π½ βΎt π) = ((TopOpenββfld) βΎt π) |
lmlim.x | β’ π β β |
Ref | Expression |
---|---|
lmlim | β’ (π β (πΉ(βπ‘βπ½)π β πΉ β π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 β’ (π½ βΎt π) = (π½ βΎt π) | |
2 | nnuz 12865 | . . 3 β’ β = (β€β₯β1) | |
3 | cnex 11191 | . . . . 5 β’ β β V | |
4 | 3 | a1i 11 | . . . 4 β’ (π β β β V) |
5 | lmlim.x | . . . . 5 β’ π β β | |
6 | 5 | a1i 11 | . . . 4 β’ (π β π β β) |
7 | 4, 6 | ssexd 5325 | . . 3 β’ (π β π β V) |
8 | lmlim.j | . . . . 5 β’ π½ β (TopOnβπ) | |
9 | 8 | topontopi 22417 | . . . 4 β’ π½ β Top |
10 | 9 | a1i 11 | . . 3 β’ (π β π½ β Top) |
11 | lmlim.p | . . 3 β’ (π β π β π) | |
12 | 1z 12592 | . . . 4 β’ 1 β β€ | |
13 | 12 | a1i 11 | . . 3 β’ (π β 1 β β€) |
14 | lmlim.f | . . 3 β’ (π β πΉ:ββΆπ) | |
15 | 1, 2, 7, 10, 11, 13, 14 | lmss 22802 | . 2 β’ (π β (πΉ(βπ‘βπ½)π β πΉ(βπ‘β(π½ βΎt π))π)) |
16 | lmlim.t | . . . . 5 β’ (π½ βΎt π) = ((TopOpenββfld) βΎt π) | |
17 | 16 | fveq2i 6895 | . . . 4 β’ (βπ‘β(π½ βΎt π)) = (βπ‘β((TopOpenββfld) βΎt π)) |
18 | 17 | breqi 5155 | . . 3 β’ (πΉ(βπ‘β(π½ βΎt π))π β πΉ(βπ‘β((TopOpenββfld) βΎt π))π) |
19 | 18 | a1i 11 | . 2 β’ (π β (πΉ(βπ‘β(π½ βΎt π))π β πΉ(βπ‘β((TopOpenββfld) βΎt π))π)) |
20 | eqid 2733 | . . . 4 β’ ((TopOpenββfld) βΎt π) = ((TopOpenββfld) βΎt π) | |
21 | eqid 2733 | . . . . . 6 β’ (TopOpenββfld) = (TopOpenββfld) | |
22 | 21 | cnfldtop 24300 | . . . . 5 β’ (TopOpenββfld) β Top |
23 | 22 | a1i 11 | . . . 4 β’ (π β (TopOpenββfld) β Top) |
24 | 20, 2, 7, 23, 11, 13, 14 | lmss 22802 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π β πΉ(βπ‘β((TopOpenββfld) βΎt π))π)) |
25 | fss 6735 | . . . . 5 β’ ((πΉ:ββΆπ β§ π β β) β πΉ:ββΆβ) | |
26 | 14, 5, 25 | sylancl 587 | . . . 4 β’ (π β πΉ:ββΆβ) |
27 | 21, 2 | lmclimf 24821 | . . . 4 β’ ((1 β β€ β§ πΉ:ββΆβ) β (πΉ(βπ‘β(TopOpenββfld))π β πΉ β π)) |
28 | 12, 26, 27 | sylancr 588 | . . 3 β’ (π β (πΉ(βπ‘β(TopOpenββfld))π β πΉ β π)) |
29 | 24, 28 | bitr3d 281 | . 2 β’ (π β (πΉ(βπ‘β((TopOpenββfld) βΎt π))π β πΉ β π)) |
30 | 15, 19, 29 | 3bitrd 305 | 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 1542 β wcel 2107 Vcvv 3475 β wss 3949 class class class wbr 5149 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcc 11108 1c1 11111 βcn 12212 β€cz 12558 β cli 15428 βΎt crest 17366 TopOpenctopn 17367 βfldccnfld 20944 Topctop 22395 TopOnctopon 22412 βπ‘clm 22730 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-lm 22733 df-xms 23826 df-ms 23827 |
This theorem is referenced by: lmlimxrge0 32928 |
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